Invariant Combinatorics on Borel Equivalence Relations Citation Wolman, Michael Solomon (2025) Invariant Combinatorics on Borel Equivalence Relations. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/w4nq-ag86. https://resolver.caltech.edu/CaltechTHESIS:06022025-192005565 Abstract This thesis comprises four independent parts and an appendix. 1. We define and study expansion problems on countable structures in the setting of descriptive combinatorics. We consider both expansions on countable Borel equivalence relations and on countable groups, in the Borel, measure, and category settings, and establish some basic correspondences between the two notions. We then explore in detail many examples, including finding spanning trees in graphs, finding monochromatic sets in Ramsey's Theorem, and linearizing partial orders. 2. Standard results in descriptive set theory provide sufficient conditions for a set P ⊆ ℕ ℕ × ℕ ℕ to admit a Borel uniformization, namely, when P has small or large sections. We consider an invariant analogue of these results with respect to a Borel equivalence relation E. Given E, we show that every such P admits an E-invariant Borel uniformization if and only if E is smooth. We also compute the definable complexity of counterexamples in the case where E is not smooth, using category, measure, and Ramsey-theoretic methods. We also show that the set of pairs (E, P) such that P has large sections and admits an E-invariant Borel uniformization is Σ 1 2 -complete. 3. Let E, F be Borel equivalence relations on X, Y, and P be an E-invariant Borel set whose sections contain countably many F-classes. We explore obstructions to the existence of Borel E-invariant uniformizing sets for P, i.e., sets choosing one F-class from every section. We survey known results, and prove new dichotomies for the case where P has σ-bounded finite sections. On the way, we prove a dichotomy characterizing the essential values of Borel cocycles into residually finite Polish groups. 4. We show that the Kechris–Solecki–Todorčević dichotomy implies the Harrington–Kechris–Louveau dichotomy. We also give a simple proof of a graph-theoretic dichotomy of Miller for doubly-indexed sequences of analytic graphs, and show that this dichotomy generalizes to finite-dimensional hypergraphs but not to ℵ 0 -dimensional hypergraphs. 5. An effective version of Nadkarni’s Theorem was proved in Ditzen’s unpublished Ph.D. thesis. The appendix contains a streamlined exposition of the proof and provides an alternative proof of the Effective Ergodic Decomposition Theorem for invariant measures (also originally proved by Ditzen). In addition, we show that the existence of an invariant Borel probability measure is not effective. Item Type: Thesis (Dissertation (Ph.D.)) Subject Keywords: descriptive set theory; logic; Borel; group; descriptive combinatorics; expansions; uniformization; effective descriptive set theory; invariant descriptive set theory; dichotomy; Borel cocycle; essential value Degree Grantor: California Institute of Technology Division: Physics, Mathematics and Astronomy Major Option: Mathematics Awards: Apostol Award for Excellence in Teaching in Mathematics, 2024. Thesis Availability: Public (worldwide access) Research Advisor(s): Kechris, Alexander S. (co-advisor) Tamuz, Omer (co-advisor) Thesis Committee: Hutchcroft, Thomas (chair) Ervin, Garrett Kechris, Alexander S. Tamuz, Omer Defense Date: 28 May 2025 Funders: Funding Agency Grant Number Fonds de Recherche du Quebec - Nature et Techonologies 290736 NSF Division of Mathematical Sciences 1950475 Record Number: CaltechTHESIS:06022025-192005565 Persistent URL: https://resolver.caltech.edu/CaltechTHESIS:06022025-192005565 DOI: 10.7907/w4nq-ag86 Related URLs: URL URL Type Description https://doi.org/10.48550/arXiv.2505.04130 arXiv Article adapted for ch. 2 https://doi.org/10.48550/arXiv.2405.15111 arXiv Article adapted for ch. 3 https://michael.wolman.ca/papers/effective_nadkarni.pdf Related Document Article adapted for appendix ORCID: Author ORCID Wolman, Michael Solomon 0009-0001-6062-4128 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 17371 Collection: CaltechTHESIS Deposited By: Michael Wolman Deposited On: 03 Jun 2025 19:24 Last Modified: 10 Jun 2025 19:56 Thesis Files PDF - Final Version See Usage Policy. 1MB Repository Staff Only: item control page

Invariant Combinatorics on Borel Equivalence Relations Thesis by Michael Solomon Wolman In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy CALIFORNIA INSTITUTE OF TECHNOLOGY Pasadena, California 2025 Defended May 28, 2025 ii © 2025 Michael Solomon Wolman ORCID: 0009-0001-6062-4128 All rights reserved iii To Esther iv ACKNOWLEDGEMENTS First and foremost I would like to thank Alexander Kechris. He has been an incredible mentor, teacher, and role model, and I am sincerely grateful for his guidance and support throughout my graduate studies. I would also like to thank Omer Tamuz and Tom Hutchcroft; they have been incredibly generous with their time and I have learned a lot from both of them. I would like to thank my mathematical colleagues and friends for many fun and interesting conversations; in particular, thank you to Clinton Conley, Frid Fu, Sita Gakkhar, Kimberly Golubeva, Jan Grebík, Patrick Lutz, Andrew Marks, Ben Miller, Marcin Sabok, Ran Tao, Asger Törnquist, Anush Tserunyan, and Robin Tucker-Drob. Thank you to Forte Shinko and Josh Frisch for their advice, support, and for being great friends, and to Garrett Ervin, Josh Frisch, Edward Hou, Forte Shinko, and Zoltán Vidnyánszky for making logic at Caltech so fun. Thank you to all of my close friends at Caltech, in Pasadena, and in LA for making my time here so special. Thank you also to Hershey and Sheva for fostering such a warm and welcoming community. Thank you to my parents for their unwavering support in all of my endeavours. 가족으로 환녕해주시고 끊임없이 사랑과 응원해주신 엄마 아빠께 감사드립니다. Thank you to the rest of my family and friends, in particular my siblings, grandparents, and B-dawg. I would also like to thank Jeff and Lesley for making LA feel like home. Thank you to Caltech for its support, both financial and otherwise, and for providing me with such a great research environment. I would like to give a special thank you to Michelle Vine in particular. I would also like to acknowledge the financial support I received from FRQNT Grant 290736 and NSF Grant DMS-1950475, which partially funded my research. Finally, thank you to Esther for encouraging me to pursue my dreams. Words cannot express how grateful I am for her endless love, support, and sacrifice, without which this would not have been possible. v ABSTRACT This thesis comprises four independent parts and an appendix. 1. We define and study expansion problems on countable structures in the setting of descriptive combinatorics. We consider both expansions on countable Borel equivalence relations and on countable groups, in the Borel, measure, and category settings, and establish some basic correspondences between the two notions. We then explore in detail many examples, including finding spanning trees in graphs, finding monochromatic sets in Ramsey’s Theorem, and linearizing partial orders. 2. Standard results in descriptive set theory provide sufficient conditions for a set P ⊆ NN × NN to admit a Borel uniformization, namely, when P has small or large sections. We consider an invariant analogue of these results with respect to a Borel equivalence relation E. Given E, we show that every such P admits an E-invariant Borel uniformization if and only if E is smooth. We also compute the definable complexity of counterexamples in the case where E is not smooth, using category, measure, and Ramsey-theoretic methods. We also show that the set of pairs (E, P ) such that P has large sections and admits an E-invariant Borel uniformization is Σ12 -complete. 3. Let E, F be Borel equivalence relations on X, Y , and P be an E-invariant Borel set whose sections contain countably many F -classes. We explore obstructions to the existence of Borel E-invariant uniformizing sets for P , i.e., sets choosing one F -class from every section. We survey known results, and prove new dichotomies for the case where P has σ-bounded finite sections. On the way, we prove a dichotomy characterizing the essential values of Borel cocycles into residually finite Polish groups. 4. We show that the Kechris–Solecki–Todorčević dichotomy implies the Harrington– Kechris–Louveau dichotomy. We also give a simple proof of a graph-theoretic dichotomy of Miller for doubly-indexed sequences of analytic graphs, and show that this dichotomy generalizes to finite-dimensional hypergraphs but not to ℵ0 -dimensional hypergraphs. 5. An effective version of Nadkarni’s Theorem was proved in Ditzen’s unpublished Ph.D. thesis. The appendix contains a streamlined exposition of the proof and provides an alternative proof of the Effective Ergodic Decomposition Theorem for invariant measures (also originally proved by Ditzen). In addition, we show that the existence of an invariant Borel probability measure is not effective. vi PUBLISHED CONTENT AND CONTRIBUTIONS [KW24a] Alexander S. Kechris and Michael S. Wolman. Ditzen’s effective version of Nadkarni’s Theorem. 2024. url: https://michael.wolman.ca/ papers/effective_nadkarni.pdf (05/05/2025). Submitted. All authors contributed equally to this project. [KW24b] Alexander S. Kechris and Michael S. Wolman. Invariant uniformization. 2024. arXiv: 2405.15111. Submitted. All authors contributed equally to this project. [Wol25] Michael S. Wolman. Definable expansions on countable groups and countable Borel equivalence relations. 2025. arXiv: 2505.04130. Preprint. This paper is entirely my own work. vii CONTENTS Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Published Content and Contributions . . . . . . . . . . . . . . . . . . . . . . . Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter I: Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Definable expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Invariant uniformization . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Invariant uniformization over quotients . . . . . . . . . . . . . . . . . 1.4 Descriptive dichotomy theorems . . . . . . . . . . . . . . . . . . . . . 1.5 Ditzen’s effective version of Nadkarni’s Theorem . . . . . . . . . . . . Chapter II: Definable expansions on countable groups and countable Borel equivalence relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Languages and structures . . . . . . . . . . . . . . . . . . . . 2.2.2 Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Equivariant and random expansions . . . . . . . . . . . . . . 2.2.4 Countable Borel equivalence relations . . . . . . . . . . . . . . 2.2.5 Structures and expansions on CBER . . . . . . . . . . . . . . 2.3 General results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Universal structurings of CBER . . . . . . . . . . . . . . . . . 2.3.2 Expansions on CBER induced by free actions . . . . . . . . . 2.3.3 Uniform random expansions . . . . . . . . . . . . . . . . . . . 2.3.4 Invariant random expansions on CBER . . . . . . . . . . . . . 2.3.5 Generic equivariant expansions . . . . . . . . . . . . . . . . . 2.3.6 Enforcing smoothness . . . . . . . . . . . . . . . . . . . . . . 2.4 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Bijections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Ramsey’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Linearizations . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.4 Vertex colourings . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.5 Spanning trees . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.6 Z-lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.7 Vizing’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.8 Matchings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.A Existence of frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter III: Invariant uniformization . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv v vi vii 1 2 5 7 9 10 12 12 18 18 19 21 22 24 24 25 27 28 29 33 41 43 45 47 50 55 57 58 61 63 67 68 72 72 viii 3.1.1 Invariant uniformization and smoothness . . . . . . . . . . . . 3.1.2 Local dichotomies . . . . . . . . . . . . . . . . . . . . . . . . 3.1.3 Anti-dichotomy results . . . . . . . . . . . . . . . . . . . . . . 3.1.4 Invariant countable uniformization . . . . . . . . . . . . . . . 3.1.5 Further invariant uniformization results and smoothness . . . 3.1.6 Remarks on invariant uniformization over products . . . . . . 3.2 Proof of Theorem 3.1.5 . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Proofs of Theorems 3.1.6 and 3.1.7 . . . . . . . . . . . . . . . . . . . . 3.4 Dichotomies and anti-dichotomies . . . . . . . . . . . . . . . . . . . . 3.4.1 Proof of Theorem 3.1.8 . . . . . . . . . . . . . . . . . . . . . . 3.4.2 An ℵ0 -dimensional (G0 , H0 ) dichotomy . . . . . . . . . . . . . 3.4.3 Proof of Theorem 3.1.8 from the ℵ0 -dimensional (G0 , H0 ) dichotomy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4.4 Proof of Theorem 3.1.10 . . . . . . . . . . . . . . . . . . . . . 3.4.5 Proofs of Proposition 3.1.11 and Theorem 3.1.12 . . . . . . . 3.4.6 Proof of Proposition 3.1.14 . . . . . . . . . . . . . . . . . . . 3.5 On Conjecture 3.1.15 . . . . . . . . . . . . . . . . . . . . . . . . . . . Chapter IV: Invariant uniformization over quotients . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Dichotomies for essential values into pro-finite groups . . . . . . . . . 4.2.1 Technical preliminaries . . . . . . . . . . . . . . . . . . . . . . 4.2.2 The G0 dichotomy for sequences of graphs . . . . . . . . . . . 4.2.3 Proof of Theorem 4.1.11 . . . . . . . . . . . . . . . . . . . . . 4.2.4 A parametrized version of Theorem 4.1.11 . . . . . . . . . . . 4.3 Proofs of Lusin–Novikov dichotomies over quotients . . . . . . . . . . Chapter V: Descriptive dichotomy theorems . . . . . . . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 A proof of the E0 dichotomy from the G0 dichotomy . . . . . . . . . . 5.3 Dichotomies for doubly-indexed sequences of graphs . . . . . . . . . . 5.4 A notion of largeness for subgraphs of G0 . . . . . . . . . . . . . . . . Appendix A: Ditzen’s effective version of Nadkarni’s Theorem . . . . . . . . . A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 A representation of ∆11 equivalence relations . . . . . . . . . . . . . . A.3 Proof of Effective Nadkarni . . . . . . . . . . . . . . . . . . . . . . . . A.4 A counterexample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5 Proof of Effective Ergodic Decomposition . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 74 76 77 78 79 80 87 89 89 93 99 100 102 105 106 110 110 118 118 122 122 127 131 138 138 142 145 148 150 150 153 156 168 170 174 1 Chapter 1 INTRODUCTION Broadly speaking, descriptive set theory is the study of definable sets in Polish spaces, i.e., separable and completely metrizable topological spaces. Examples of Polish spaces include the space R of real numbers, the Cantor space 2N of infinite binary sequences, and the Baire space NN of infinite sequences of natural numbers. What it means to be definable is intentionally flexible; it is beneficial to consider various notions of definability depending on the context. Most commonly we consider the Borel sets, that is, the sets generated by the basic open sets via the operations of countable union, countable intersection and complementation. In this thesis we also consider various other classes of sets, such as the larger classes of analytic, measurable, or Baire measurable sets, or the more restricted classes of Gδ sets or “effectively Borel” sets. Restricting ourselves to definable sets has many benefits. First, we avoid pathologies that may exist in the general context, such as non-measurable sets or sets of reals of cardinality strictly between ℵ0 and the continuum. Second, using the structure afforded to us by the definitions of these sets, we get more refined and broadly applicable versions of general results. For example, the Perfect Set Theorem asserts that not only does the continuum hypothesis hold for analytic sets, but in this case there is a concrete witness to uncountability—every uncountable analytic set contains a homeomorphic copy of the Cantor space. As another example, by the Lusin–Novikov Uniformization Theorem not only do Borel families of countable sets have choice functions (a consequence of the axiom of choice), such families admit choice functions that are Borel. Finally, by considering the definable complexity of sets and functions between them, we get a much finer analysis of and comparison between the complexity of various sets, structures, and combinatorial problems; one can view this as analogous to the study of complexity and reductions in computer science. One concept that has become especially important, both in theory and in application, is that of a Borel equivalence relation. A Borel equivalence relation on a Polish space X is an equivalence relation E that is Borel, considered as a set of pairs in the product X × X. The main notion of relative complexity between various Borel equivalence relations is that of Borel reduction: if E is a Borel equivalence relation on X and F is a Borel equivalence relation on Y , a Borel reduction from E to F is a Borel 2 function f : X → Y so that x0 Ex1 ⇐⇒ f (x0 )F f (x1 ). We write E ≤B F when there is a Borel reduction from E to F , and in this case we view F as being “at least as complicated” as E. We give a brief overview of the “base” of the poset of Borel equivalence relations with respect to Borel reducibility: The simplest Borel equivalence relations are the smooth ones, i.e., those that admit Borel reductions to equality on R. Above these there is E0 , the eventual equality relation on 2N : xE0 y ⇐⇒ ∃n∀k ≥ n(xk = yk ). By the Harrington–Kechris–Louveau Theorem, if E is a Borel equivalence relation that is not smooth, then E0 ≤B E. The structure of the Borel equivalence relations above E0 is much more complicated; see for example [Gao08; Kan08; Kec25]. The central focus of this thesis is on finding Borel solutions to combinatorial problems that are “compatible with” or “invariant with respect to” various Borel equivalence relations. We explain in detail below what this means in various contexts, and give an overview of our results. 1.1 Definable expansions A countable Borel equivalence relation is an equivalence relation whose equivalence classes are countable. Given a countable Borel equivalence relation E, a Borel structuring of E is a Borel assignment of a first-order structure on each E-class C. When each of these structures comes from a class K of countable structures, we call this a Borel K-structuring of E. Broadly speaking, given a combinatorial problem on countable structures, we are interested in solving it in a “uniformly Borel” way on a countable Borel equivalence relation, possibly after throwing away a meagre set or a null set. There are various examples of interest coming from graph theory, such as finding (edge) colourings, perfect matchings or spanning trees in graphs. Other examples include finding infinite monochromatic sets as in Ramsey’s Theorem, or linearizing partial orders. Such problems have been studied extensively in this context; see for example [KM20; Pik21; CK18; GX24]. In Chapter 2, we consider these problems within the framework of expansions. Given first-order languages L ⊆ L∗ and an L-structure A, we call an L∗ -structure A∗ an expansion of A if A = A∗ L, where A∗ L denotes the reduct of A∗ to L. If K is a class of L structures and K∗ is a class of L∗ -structures, the expansion problem for (K, K∗ ) 3 is the problem of determining whether every structure in K admits an expansion in K∗ . All of the combinatorial problems above, such as graph colourings and Ramsey’s Theorem, can be phrased as expansion problems. We study here the “uniformly Borel” analogue of expansion problems. That is, we consider expansion problems (K, K∗ ) for which every element of K admits an expansion to an element of K∗ . Given such a problem, a countable Borel equivalence relation E and a Borel K-structuring A of E, we study the following problem: is there a Borel K∗ -structuring A∗ of E whose restriction to every E-class is an expansion of the original structuring A? We also study an “equivariant” version of the Borel expansion problem. Let Γ be a group and (K, K∗ ) is an expansion problem with K, K∗ Borel. We let K(Γ) (resp. K∗ (Γ)) denote the set of structures in K (resp. K∗ ) whose universe is Γ. The action of Γ on itself by multiplication on the left induces a natural action of Γ on K(Γ), K∗ (Γ). The Γ-equivariant expansion problem is then the following: is there a Borel map f : K(Γ) → K∗ (Γ), taking A ∈ K(Γ) to an expansion f (A), which is equivariant with respect to the induced action of Γ on K(Γ), K∗ (Γ)? Our primary objective is to study this correspondence between the Borel expansion problem on CBER and the Borel equivariant expansion for countable groups Γ. More generally, we study also the connection between these problems in the settings of measure and category, i.e., when we are allowed to solve these problems after possibly removing a null or meagre set. We show that there is natural correspondence between Γ-equivariant expansions for countable groups Γ and countable Borel equivalence relations which arise via free Borel actions of Γ. We then apply our results to various examples. We include below a representative sample of our results; see Chapter 2 for more precise definitions and further results. In terms of measure, we consider random expansions for countable groups, where we say an invariant measure ν on K∗ (Γ) is a random expansion of an invariant measure µ on K(Γ) when the reduct of ν is equal to µ. We show that the existence of random expansions on Γ depends only on its orbit equivalence class, where we say groups Γ, Λ are orbit equivalent if there is a countable Borel equivalence relation E induced by free probability-measure-preserving actions of both Γ and Λ. Theorem 1.1.1. Let (K, K∗ ) be an expansion problem and Γ, Λ be countably infinite groups. If Γ, Λ are orbit equivalent, then Γ admits random expansions from K to K∗ if 4 and only if Λ admits random expansions from K to K∗ . For category, we consider generic equivariant expansions on Gδ classes of structures, i.e., equivariant expansions on comeagre subsets of K(Γ). Given an expansion problem (K, K∗ ) and a countably infinite group Γ, we say K admits Γ-equivariant expansions generically if there is a comeagre invariant Borel set X ⊆ K(Γ) such that there is a Borel Γ-equivariant expansion map X → K∗ (Γ). We show that when K consists of structures with trivial algebraic closure that are not definable from equality, whether or not K admits Γ-equivariant expansions to K∗ generically is independent of the group Γ. (A structure is said to have trivial algebraic closure if its automorphism group has infinite orbits, even after fixing finitely many points, and is definable from equality when relations between tuples of points depend only on their equality types.) Theorem 1.1.2. Let (K, K∗ ) be an expansion problem. Suppose that K is Gδ and the generic element of K has trivial algebraic closure and is not definable from equality. Then the following are equivalent: 1. For every countably infinite group Γ, K admits Γ-equivariant expansions to K∗ generically. 2. There exists a countably infinite group Γ for which K admits Γ-equivariant expansions to K∗ generically. As one concrete example, we consider the problem of choosing from a countable linear order without endpoints a subset that is order-isomorphic to Z, in a Borel way. Theorem 1.1.3. 1. Let K be the class of linear orders without endpoints, and K∗ be the class of linear orders without endpoints along with a subset of order-type Z. For any countably infinite group Γ, K does not admit Γ-equivariant expansions to K∗ generically. In particular, every non-smooth CBER E admits a Borel assignment of linear orders to every E-class so that there is no Borel way to choose an infinite subset of each E-class that has order-type Z. 2. There is a Borel Γ-invariant set X ⊆ K(Γ) and a Borel equivariant expansion map f : X → K∗ (Γ) such that, for all invariant random K-structures µ on Γ, µ 5 admits a random expansion to K∗ if and only if µ(X) = 1, in which case f∗ µ gives such an expansion. Moreover, we can choose f so that for all L ∈ X, f (L) picks out an interval in L. 1.2 Invariant uniformization Let X, Y be sets and let P ⊆ X ×Y satisfy ∀x ∈ X∃y ∈ Y (x, y) ∈ P . A uniformization of P is a function f : X → Y so that ∀x ∈ X((x, f (x)) ∈ P ). If E is an equivalence relation on X, we say P is E-invariant if x0 Ex1 =⇒ Px0 = Px1 , where Px = {y ∈ Y : (x, y) ∈ P } is the x-section of P . In this case, an E-invariant uniformization is a uniformization f such that x0 Ex1 =⇒ f (x0 ) = f (x1 ). A uniformization for P can be viewed as a choice function for the family {Px : x ∈ X}. When X, Y are Polish spaces and P is Borel, standard results in descriptive set theory give sufficient conditions for the existence of Borel uniformizations of P , such as when P has small or large sections; see e.g. [Kec95, Section 18]. Suppose now that P has small or large sections. In Chapter 3, we study the existence of Borel E-invariant uniformizations of P , when E is a Borel equivalence relation on X and P is E-invariant. Given a Borel equivalence relation E, we consider the property that every E-invariant P with countable (resp. Kσ , non-meagre, non-null) sections admits a Borel E-invariant uniformization. We show that in every case, E has this property if and only if E is smooth. Theorem 1.2.1 (Kechris–Wolman). Let E be a Borel equivalence relation on a Polish space X. Then the following are equivalent: (i) E is smooth; (ii) every E-invariant Borel set P with non-null sections admits a Borel E-invariant uniformization; (iii) every E-invariant Borel set P with non-meagre sections admits a Borel Einvariant uniformization; (iv) every E-invariant Borel set P with Kσ sections admits a Borel E-invariant uniformization; (v) every E-invariant Borel set P with countable sections admits a Borel E-invariant uniformization. 6 We also characterize the minimal definable complexity of counterexamples. Below we state the largest class for which there are always Borel invariant uniformizations; for each of these cases, we also give counterexamples at the next level of the Borel hierarchy. Theorem 1.2.2 (Kechris–Wolman). Let X, Y be Polish spaces, E a Borel equivalence relation on X, and P ⊆ X × Y an E-invariant Borel relation. Suppose one of the following holds: (i) Px ∈ ∆02 and µx (Px ) > 0, for all x ∈ X, and some Borel assignment x 7→ µx of probability Borel measures µx on Y ; (ii) Px ∈ Fσ and Px non-meager, for all x ∈ X; (iii) Px ∈ Gδ and Px non-empty and Kσ (in particular countable), for all x ∈ X. Then there is a Borel E-invariant uniformization. We next consider “local” dichotomies. Miller recently proved a dichotomy showing that E0 is essentially the only obstruction to the existence of invariant uniformizations, in the case of countable sections [Mild, Theorem 2]. We provide a different proof of this dichotomy, using Miller’s (G0 , H0 ) dichotomy [Mil12] and Lecomte’s ℵ0 -dimensional hypergraph dichotomy [Lec09]. We also prove an ℵ0 -dimensional (G0 , H0 )-type dichotomy, which generalizes Lecomte’s dichotomy in the same way that the (G0 , H0 ) dichotomy generalizes the G0 dichotomy, and use this to give still another proof of this theorem. Informally, dichotomies such as [Mild, Theorem 2] provide upper bounds on the complexity of the collection of Borel sets satisfying certain combinatorial properties; for example, (the effective version of) Miller’s dichotomy gives a bound of Π11 for the set of pairs (E, P ) admitting Borel E-invariant uniformizations, when P has countable sections. Thus, one method of showing that there is no analogous dichotomy in other cases is to provide lower bounds on the complexity of such sets. We show that this is the case for the large section problem, namely, we show that the set of such pairs (E, P ), where P has large sections, is Σ12 -complete. Theorem 1.2.3 (Kechris–Wolman). Let P be the class of pairs (E, P ) such that E is a Borel equivalence relation on NN and P ⊆ NN × NN is Borel and E-invariant, has large sections, and admits an E-invariant Borel uniformization. Then P is Σ12 -complete. 7 The following is still open: Problem 1.2.4. Is there an analogous dichotomy or anti-dichotomy result for the case where P has Kσ sections? We end the chapter with some partial results concerning the more general problem of invariant countable uniformization, i.e., invariant uniformization where we choose a countable set in each section instead of a single point. 1.3 Invariant uniformization over quotients Suppose now that E is a Borel equivalence relation on a Polish space X and F is a Borel equivalence relation on a Polish space Y . We say P ⊆ X × Y is E × F -invariant if x0 Ex1 & y0 F y1 & (x0 , y0 ) ∈ P =⇒ (x1 , y1 ) ∈ P. Suppose now that P is Borel and E × F -invariant. We say P has countable sections over F if the sections of P each contain countably many F -classes. A Borel E-invariant uniformization of P over F is a Borel set U ⊆ P that is E × F -invariant, and whose sections each contain exactly one F -class. In this chapter, we look at dichotomies characterizing the existence of Borel E-invariant uniformizations of P over F , when P is Borel, E × F -invariant and has countable sections over F . We begin by considering the case where the sections of P contain finitely many F -classes. The following has been shown independently by the author and Miller (personal communication), who pointed out that it follows from their work on essential values of Borel cocycles. Theorem 1.3.1. Let E, F be Borel equivalence relations on Polish spaces X, Y , and P ⊆ X × Y be a Borel E × F -invariant set whose sections contain exactly n F -classes, n ≥ 2. There is a finite basis of minimal obstructions to the existence of Borel E-invariant uniformizations of P over F , corresponding to the set of minimal fixed-point-free subgroups of Sn (up to conjugacy). If E is an equivalence relation and Γ is a group, a map ρ : E → Γ is a cocycle if it satisfies the cocycle identity ρ(x, y)ρ(y, z) = ρ(x, z) 8 for xEyEx. If Γ is a countable discrete group, E is a Borel equivalence relation on a Polish space X, ρ : E → Γ is a Borel cocycle and Λ ⊆ Γ is a non-empty set, we say Λ is an essential value of ρ if for every partition (Bn )n∈N of X into Borel sets, there is some n so that for all λ ∈ Λ there are x 6= y ∈ Bn with ρ(x, y) = λ. If Γ is a group, E is a Borel equivalence relation on X, F is a Borel equivalence relation on Y , and ρ : E → Γ, π : F → Γ are cocycles, a continuous embedding of ρ into π is a continuous injection f : X → Y so that x0 Ex1 ⇐⇒ f (x0 )F f (x1 ), and ρ(x0 , x1 ) = π(f (x0 ), f (x1 )) for x0 Ex1 . Miller has proved the following dichotomy characterizing the essential values of Borel cocycles into countable groups. Theorem 1.3.2 (Miller [Mila, Theorem 1]). Suppose Λ ≤ Γ are countable discrete non-trivial groups. There is a canonical cocycle ρΛ : E0 → Λ so that for every Borel equivalence relation E and every Borel cocycle ρ : E → Γ, the following are equivalent: 1. Λ is an essential value of ρ. 2. There is a continuous embedding of ρΛ into ρ. We include a simple proof of Theorem 1.3.1 from Miller’s dichotomy for essential values of Borel cocycles. We then consider the case where P has countable sections over F , but can be partitioned into countably many E × F -invariant Borel sets that have bounded finite sections over F . We prove a dichotomy characterizing the essential values of Borel cocycles into pro-finite Polish groups, and use this to generalize Theorem 1.3.1 to this setting. Theorem 1.3.3. Let (Γn )n∈N be a sequence of finite groups, and Fn be a family of non-trivial subsets of Γn that is closed under conjugation. For every Borel equivalence Q relation E on X and every Borel cocycle ρ : E → n Γn , the following are equivalent: 1. The family (Fn )n is an essential value for ρ, meaning that for every cover of X by Borel sets Bi,k , there are i, k such that projΓi (ρ(EBi,k \ ∆(Bi,k ))) contains an element of Fi . 2. There is a subgroup Λ ≤ n Γn so that projΓn (Λ) contains an element of Fn for Q all n ∈ N, a (somewhat canonical) cocycle ρ : E0 → n Γn that has Λ as an essential value, and a continuous embedding of ρ into ρ. Q 9 Theorem 1.3.4. Let E, F be Borel equivalence relations on Polish spaces X, Y , and Pn ⊆ X × Y be Borel E × F -invariant sets whose sections contain exactly α(n) ≥ 2 F -classes. There is a (somewhat canonical) basis of minimal obstructions to the S existence of Borel E-invariant uniformizations of n Pn over F , corresponding to Q fixed-point-free subgroups of n Sα(n) . 1.4 Descriptive dichotomy theorems We have discussed so far various dichotomy theorems in descriptive set theory concerning Borel equivalence relations. We recall now some of the most important, namely Silver’s Theorem and the Harrington–Kechris–Louveau Theorem. If E is a Borel equivalence relation on X and F is a Borel equivalence relation on Y , a continuous embedding of E into F is a reduction f : X → Y from E to F that is continuous and injective. Theorem 1.4.1 (Silver). Let E be a co-analytic equivalence relation on a Polish space. Exactly one of the following hold: 1. E has countably many equivalence classes. 2. There is a continuous embedding of equality on 2N into E. Theorem 1.4.2 (Harrington–Kechris–Louveau). Let E be a Borel equivalence relation on a Polish space. Exactly one of the following hold: 1. E is smooth, i.e., there is a Borel reduction of E to equality on R. 2. There is a continuous embedding of E0 into E. We note that the Harrington–Kechris–Louveau Theorem extends prior results of Glimm and Effros, who proved this in the case that E is induced by a continuous action of a locally compact Polish group (and in particular when E is countable). We also recall the graph-theoretic dichotomy of Kechris–Solecki–Todorčević. We say a (directed) graph G on a Polish space X is analytic if it is analytic as a subset of X × X, and we say G has countable Borel chromatic number if there is a cover of X by countably many G-independent Borel sets. If G is a graph on X and H is a graph on Y , a homomorphism from G to H is a map f : X → Y so that x0 Gx1 =⇒ f (x0 )Hf (x1 ). 10 Fix a set S ⊆ 2<N that contains exactly one sequence of every length, and which is dense, meaning that for all t ∈ 2<N there is some s ∈ S with t ⊆ s. Define the directed graph G0 on 2N by G0 = {(s_ (0)_ x, s_ (1)_ x) : s ∈ S, x ∈ 2N }, where _ denotes concatenation of sequences. Theorem 1.4.3 (Kechris–Solecki–Todorčević). Let G be a directed analytic graph on a Polish space. Exactly one of the following hold: 1. G has countable Borel chromatic number. 2. There is a continuous homomorphism of G0 into G. The original proofs of all of these dichotomies used techniques of effective descriptive set theory. In [Mil12], Miller found a classical proof of the Kechris–Solecki–Todorčević Theorem, and used this to give a classical proof of Silver’s Theorem and the Glimm– Effros dichotomy for countable Borel equivalence relations. Miller also proved a generalization of the Kechris–Solecki–Todorčević Theorem, and used this to give a classical proof of the full Harrington–Kechris–Louveau Theorem. Since then, it has remained an open question whether there is a proof of the Harrington– Kechris–Louveau Theorem directly from the Kechris–Solecki–Todorčević Theorem. In Chapter 5, we show that this is indeed the case. Theorem 1.4.4. There is a proof of the Harrington–Kechris–Louveau Theorem directly from the Kechris–Solecki–Todorčević Theorem. We then give new proofs of a dichotomy of Miller for doubly-indexed sequences of analytic graphs and show that this dichotomy generalizes to finite-dimensional hypergraphs but not infinite-dimensional hypergraphs, in contrast to most other graph-theoretic dichotomies. We also prove that if G0 is written as a finite union of Baire measurable graphs, then there is a continuous embedding of G0 into one of these graphs, without applying the Kechris–Solecki–Todorčević Theorem. 1.5 Ditzen’s effective version of Nadkarni’s Theorem An effective version of Nadkarni’s Theorem was proved in Ditzen’s unpublished Ph.D. thesis. Appendix A contains a streamlined exposition of the proof, and provides 11 an alternative proof of the Effective Ergodic Decomposition Theorem for invariant measures (also originally proved by Ditzen). In addition, we show that the existence of an invariant Borel probability measure is not effective. We use this example to construct an effectively Borel non-smooth equivalence relation that does not effectively admit a compact action realization, giving a concrete witness to [FKSV23, Proposition 4.3.17]. 12 Chapter 2 DEFINABLE EXPANSIONS ON COUNTABLE GROUPS AND COUNTABLE BOREL EQUIVALENCE RELATIONS Michael S. Wolman 2.1 Introduction A countable Borel equivalence relation (CBER) on a Polish space X is a Borel equivalence relation E ⊆ X 2 whose equivalence classes are countable. Given a CBER E, a (Borel) structuring of E is a Borel assignment of a first-order structure on each E-class C (see Sections 2.2.5 and 2.3.1 for precise definitions). In this paper, we are primarily concerned with the descriptive combinatorics of locally countable structures. Broadly speaking, given a combinatorial problem on countable structures, we are interested in solving it in a “uniformly Borel” way, possibly after throwing away a meagre set or a null set. For instance, given a Borel structuring of a CBER E by countable graphs, we may be interested in characterizing exactly when one can find a Borel colouring of these graphs with countably many colours, i.e., a colouring so that the assignment of the colour classes to the vertices in each E-class is a Borel structuring of E. Other examples of combinatorial problems include finding proper edge colourings, perfect matchings or spanning trees in graphs, finding infinite monochromatic sets (as in Ramsey’s Theorem), and extending a given partial order into a linear order; see Section 2.2.2 for more. We refer the reader to [KM20; Pik21] for a survey of results in descriptive combinatorics, and to [CK18; BC24] for more on the structurability of CBER. For “locally finite” structures, many of these combinatorial problems can be expressed in terms of constrain satisfaction or locally checkable labelling problems on graphs. In this setting, there has been a lot of recent progress towards finding solutions to various expansion problems, using tools from theoretical computer science and finite combinatorics such as the Lovász Local Lemma and connections with LOCAL algorithms in distributed computing [Ber23; BCGGRV22; GR23]. However, we note that many problems are not locally finite and hence do not fit within this framework; for example, linearizations of partial orders, or Ramsey’s Theorem (see e.g. [GX24]). 13 Here, we consider these problems in the more general framework of expansions. Given first-order languages L ⊆ L∗ and an L-structure A, we call an L∗ -structure A∗ an expansion of A if A = A∗ L, where A∗ L denotes the reduct of A∗ to L. If K is a class of L structures and K∗ is a class of L∗ -structures, the expansion problem for (K, K∗ ) is the problem of determining whether every structure in K admits an expansion in K∗ . For a countably infinite set X, let K(X) denote the set of structures in K whose universe is X, and call K a Borel class of structures if K(X) is Borel for all countably infinite sets X. Given an expansion problem (K, K∗ ) for which every element of K admits an expansion to an element of K∗ , we get a corresponding “uniformly Borel” expansion problem: For every CBER E and any structuring of E with elements of K, is there a structuring of E with elements of K∗ which is an expansion of the original structuring on every E-class? In general, one can view this Borel expansion problem as asking if there is a “canonical” assignment of an expansion in K∗ to every element of K; this is made precise in [CK18, arXiv version, Appendix B; BC24]. One can also interpret the Borel expansion problem in terms of definable equivariant maps. If Γ is a group acting on a countably infinite set X and (K, K∗ ) is an expansion problem with K, K∗ Borel, we may consider the Γ-equivariant expansion problem: Is there a Borel map f : K(X) → K∗ (X), taking A ∈ K(X) to an expansion f (A), which is equivariant with respect to the induced action of Γ on K(X), K∗ (X)? There is a natural correspondence between Γ-equivariant expansions for countable groups Γ, in the special case where Γ acts on X = Γ by multiplication on the left, and CBER which arise via free Borel actions of Γ (see Section 2.3.2). By the Feldman–Moore Theorem, every CBER is induced by a Borel action of a countable group, though in general we cannot expect this action to be free [Kec25, Section 11]. Nevertheless, CBER induced by free Borel actions of countable groups are a great source of (counter-)examples in the study of Borel expansions on CBER (especially with respect to the Schreier graphs of their actions), and remain very relevant in the study of the descriptive combinatorics of locally countable structures. The primary objective of this paper is to study this correspondence between the Borel expansion problem on CBER and the Borel equivariant expansion for countable groups Γ. More generally, we study also the connection between these problems in the settings of measure and category, i.e., when we are allowed to solve these problems after possibly removing a null or meagre set. We shall see that by exploiting this connection, we can apply results and techniques from the theory of CBER to prove theorems about 14 equivariant expansions on countable groups (see e.g. Sections 2.3.4, 2.3.5, and 2.4). Conversely, we apply tools from symbolic dynamics and probability theory (such as the mass transport principle and random walks on groups) to the study of equivariant expansions, which gives in some cases precise characterizations of exactly when certain structurings of CBER admit definable expansions (c.f. Sections 2.3.3, 2.3.6, and 2.4). The connection between expansions on CBER and equivariant expansions in the purely Borel setting has been studied independently in [BC24], with the goal of making precise the relation between the existence of Borel expansions on CBER and “canonical” expansions from K to K∗ . There, Banerjee and Chen [BC24, Corollary 3.29, Remark 2.25] show that every Borel structuring of a CBER admits a Borel expansion if and only if there is a Borel SN -equivariant expansion map, for classes K of structures that interpret the theories of Lusin–Novikov functions and countable separating families (c.f. [BC24, Definitions 3.18, 3.23]). (We note however that the classes we study in this paper do not interpret these theories.) Expansion problems have also been studied in the context of invariant random structures on groups, i.e., invariant probability measures on K(Γ), and one can view equivariant expansions as a natural strengthening of this notion; see e.g. [KM20, Sections 6, 15] for some examples in graph combinatorics, or [GLM24; Alp22] for linearizations of partial orders. Organization. The structure of this paper is as follows. In Section 2.2 we give precise definitions of expansions on CBER and equivariant expansions on groups for expansion problems, in the Borel, Baire category, and measurable settings. We also give examples of various expansion problems of interest, that we study in detail in Section 2.4. In Section 2.3, we prove several general theorems relating equivariant expansions on groups Γ with Borel expansions on CBER induced by free Borel actions of Γ. We describe in Section 2.3.2 a weak duality between the two notions, which can be viewed as an analogue of [BC24, Corollary 3.29] for this setting. We then consider random expansions for countable groups, where we say an invariant measure ν on K∗ (Γ) is a random expansion of an invariant measure µ on K(Γ) when the reduct of ν is equal to µ (c.f. Section 2.2.3). We show that the existence of random expansions on Γ depends only on its orbit equivalence class, where we say groups Γ, Λ are orbit equivalent if there is a CBER E induced by free probability-measure-preserving actions of both Γ and Λ. Theorem 2.1.1 (Theorem 2.3.7). Let (K, K∗ ) be an expansion problem and Γ, Λ be 15 countably infinite groups. If Γ, Λ are orbit equivalent, then Γ admits random expansions from K to K∗ if and only if Λ admits random expansions from K to K∗ . We note that this has already been observed in some special cases, for example with linearizations in [Alp22], though we show here that it holds more generally for all expansion problems. We also give a sort of converse in Proposition 2.3.8. Next, we consider generic equivariant expansions on Gδ classes of structures, i.e., equivariant expansions on comeagre subsets of K(Γ) (c.f. Section 2.2.3). Given an expansion problem (K, K∗ ) and a countably infinite group Γ, we say K admits Γequivariant expansions generically if there is a comeagre invariant Borel set X ⊆ K(Γ) such that there is a Borel Γ-equivariant expansion map X → K∗ (Γ). We show that when K consists of structures with trivial algebraic closure that are not definable from equality, whether or not K admits Γ-equivariant expansions to K∗ generically is independent of the group Γ. (A structure is said to have trivial algebraic closure if its automorphism group has infinite orbits, even after fixing finitely many points, and is definable from equality when relations between tuples of points depend only on their equality types; see Definition 2.3.11 for precise definitions of these terms.) Theorem 2.1.2 (Theorem 2.3.13). Let (K, K∗ ) be an expansion problem. Suppose that K is Gδ and the generic element of K has trivial algebraic closure and is not definable from equality. Then the following are equivalent: 1. For every countably infinite group Γ, K admits Γ-equivariant expansions to K∗ generically. 2. There exists a countably infinite group Γ for which K admits Γ-equivariant expansions to K∗ generically. A CBER E is smooth if there is a Borel set that contains exactly one point from every E-class. We give in Section 2.3.6 sufficient conditions for an expansion problem to satisfy (a) that every structuring of a smooth CBER admits a Borel expansion (Proposition 2.3.22 and Remark 2.3.23), or (b) that every non-smooth CBER admits a structuring with no Borel expansion (Proposition 2.3.25 and Corollary 2.3.26). In Section 2.4 we analyze in detail the expansion problem for the examples described in Section 2.2.2, using in particular the tools we developed in Section 2.3. We summarize our results in Table 2.1; we highlight a few of these below. 16 Call a CBER aperiodic if it has only infinite equivalence classes. In [KST99] it is shown that for every non-smooth aperiodic CBER E, there are Borel sets A, B which have infinite intersection with every E-class, but for which there is no Borel bijection f : A → B whose graph is contained in E. By contrast, we have the following: Theorem 2.1.3 (Generic bijections (Theorem 2.4.3)). Let E be an aperiodic CBER on X and A, B ⊆ X be sets that have infinite intersection with every E-class. Then there is a comeagre E-invariant set Y ⊆ X and a Borel bijection f : A ∩ Y → B ∩ Y whose graph is contained in E, i.e., such that xEf (x) for all x ∈ A ∩ Y . The problem of whether an invariant random partial order on a countably infinite group Γ can be linearized was studied in [GLM24; Alp22]. Alpeev [Alp22] has shown that this random expansion property holds for Γ if and only if Γ is amenable. By contrast, for equivariant maps and CBER we have the following: Theorem 2.1.4 (Linearizations (Theorems 2.4.11 and 2.4.12)). 1. Let K be the class of partial orders and K∗ be the class of linear orders extending a given partial order. For every countably infinite group Γ, K does not admit Γ-equivariant expansions to K∗ generically. 2. For every non-smooth CBER E, there is a Borel assignment of a partial order to every E-class so that there is no Borel way of extending these partial orders to linear orders on every E-class. Moreover, if E is aperiodic then one can ensure that for every E-invariant probability Borel measure µ, there is no Borel extension of the partial orders to linear orders µ-a.e. A CBER E is treeable if there is a Borel assignment of a connected acyclic graph to every E-class. The class of treeable CBER has been studied extensively; see e.g. [Kec25, Section 10]. In Section 2.4.5, we consider CBER E that admit Borel spanning trees for every Borel assignment of a connected graph to every E-class. Clearly every such CBER is treeable. We show that the hyperfinite CBER have this property, where a CBER is said to be hyperfinite if it can be written as an increasing union of CBER with finite equivalence classes. Theorem 2.1.5 (Spanning trees (Theorem 2.4.16)). Let E be a CBER. If E is hyperfinite, then for every Borel assignment of a connected graph to each E-class, there is a Borel assignment of a spanning tree to each E-class. 17 It is unknown whether the class of CBER with this spanning tree property coincides with the treeable CBER or the hyperfinite CBER, or if it lies somewhere in between. As a final example, we consider the problem of choosing from a linear order without endpoints a subset that is order-isomorphic to Z, in a Borel way. We give a complete classification of the invariant random structures on countably infinite groups that admit random expansions for this problem, and show moreover that these expansions can always be taken to come from equivariant Borel maps. In particular, this characterizes exactly when a Borel structuring of a CBER admits a Borel expansion for this problem µ-a.e., for any invariant measure µ. Theorem 2.1.6 (Z-lines (Theorems 2.4.18 and 2.4.20)). 1. Let K be the class of linear orders without endpoints, and K∗ be the class of linear orders without endpoints along with a subset of order-type Z. For any countably infinite group Γ, K does not admit Γ-equivariant expansions to K∗ generically. In particular, every non-smooth CBER E admits a Borel assignment of linear orders to every E-class so that there is no Borel way to choose an infinite subset of each E-class that has order-type Z. 2. There is a Borel Γ-invariant set X ⊆ K(Γ) and a Borel equivariant expansion map f : X → K∗ (Γ) such that, for all invariant random K-structures µ on Γ, µ admits a random expansion to K∗ if and only if µ(X) = 1, in which case f∗ µ gives such an expansion. Moreover, we can choose f so that for all L ∈ X, f (L) picks out an interval in L. We also give in Section 2.4.7 a survey of recent results regarding the existence of Borel proper edge colourings of bounded-degree graphs (i.e. definable Vizing’s Theorem), and in Section 2.4.8 a survey of the current landscape regarding the existence of Borel perfect matchings in bipartite graphs (i.e. definable Hall’s Theorem). We end with a list of open problems in Section 2.5. Acknowledgements. This research was partially supported by NSF Grant DMS1950475. We would like to thank Alexander Kechris for the initial discussion that inspired this project, and Alexander Kechris, Tom Hutchcroft, Omer Tamuz, Garrett Ervin, Edward Hou, and Sita Gakkhar for many helpful conversations throughout. We thank Tom Hutchcroft for telling us about Lemma 2.4.19, and Minghao Pan for sharing with us his notes on the proof. We also thank Marcin Sabok for their many 18 comments and suggestions regarding our survey in Section 2.4.8. Finally, thank you to Esther Nam for the encouragement and support. 2.2 Preliminaries For a background on general descriptive set theory, see [Kec95]. For a survey of the theory of CBER, see [Kec25]. For the basics of structurability of CBER, see [CK18]. 2.2.1 Languages and structures By a language, we will always mean a countable relational first-order language, i.e., a countable set L = {Ri : i ∈ I}, where each Ri is a relation symbol with associated arity ni ≥ 1. Fix now a language L and let X be a set. An L-structure on X is a tuple A = (X, RA )R∈L where RA ⊆ X n for each n-ary relation symbol R ∈ L. We call X the universe of A, and let ModL (X) denote the space of L-structures on X. For A ∈ ModL (X) and Y ⊆ X, let AY ∈ ModL (Y ) denote the restriction of A to Y , given by RAY (y1 , . . . , yn ) ⇐⇒ RA (y1 , . . . , yn ) for all n-ary R ∈ L and y1 , . . . , yn ∈ Y . For L0 ⊆ L, we let AL0 = (X, RA )R∈L0 denote the reduct of A to L0 , i.e., the structure we get when we “forget” the relations in L \ L0 . (Note that the notation A(−) is used both for restrictions and reducts; which one we are referring to throughout this paper will be clear from context.) If A is an L-structure on X and f : X → Y is a bijection, we write f (A) for the push-forward structure on Y , i.e., the structure on Y given by RA (x0 , . . . , xn−1 ) ⇐⇒ Rf (A) (f (y0 ), . . . , f (yn−1 )) for all n-ary relations R ∈ L. When X = Y this defines the logic action of SX on ModL (X), where SX is the group of bijections of X. We are primarily interested in the cases where X is a countably infinite set, or when X is a Polish space. We will reserve the symbols A, B, . . . for L-structures on countable sets, and the symbols A, B, . . . for L-structures on Polish spaces. When X is a countable set, one can view ModL (X) as a compact Polish space, namely, ModL (X) = Y n 2X i . i∈I The logic action of SX on ModL (X) is continuous for this topology. 19 By a class of L-structures, we mean a class K of countably infinite L-structures closed under isomorphism. Given such a class K and a countably infinite set X, we let K(X) = K ∩ ModL (X) denote the space of L-structures in K with universe X. We call K a Borel class of L-structures (resp. a Gδ class of L-structures, a closed class of L-structures) if K(X) is Borel (resp. Gδ , closed) as a subset of ModL (X) for some (equivalently any) countably infinite set X. For a countably infinite set X and an L-structure A let AgeX (A) denote the age of A (on X), that is, the set finite L0 -structures that that embed into A and whose universe is contained in X, for finite L0 ⊆ L. Let AgeX (K) = [ {AgeX (A) : A ∈ K(X)} for any class K of L-structures. For A0 ∈ AgeX (ModL ) and A ∈ ModL (X), we write A0 v A if (AL0 )F = A0 , where A0 is an L0 -structure with universe F ⊆ X. The topology of K(X) is generated by basic clopen sets of the form N (A0 ) = {A ∈ K(X) : A0 v A} for A0 ∈ AgeX (K). We note that there is an analogous logic action of SX on AgeX (K). We note that our definition of the age of A differs from the usual one (see e.g. [Hod93, Section 7]), which considers all finite L-structures that embed into A without restricting to finite sublanguages. We choose here to restrict to finite sublanguages so that AgeX (K) corresponds naturally to a basis for the topology on K(X). (We specify the universe X in AgeX (K) for the same reason.) 2.2.2 Expansions Let L ⊆ L∗ be languages, A be an L-structure and A∗ be an L∗ -structure. We say A∗ is an expansion of A if A∗ L = A. Given a class of L-structures K and a class of L∗ -structures K∗ with L ⊆ L∗ , the expansion problem for (K, K∗ ) is the question of whether or not every element of K admits an expansion in K∗ . We call such pairs (K, K∗ ) expansion problems. Below we give examples of expansion problems (K, K∗ ) we will consider in this paper. In all of these examples the expansion problem will have a positive solution, i.e., every element of K admits an expansion in K∗ ; we will be interested in finding “definable” expansions, for various notions of definability that we make precise below. We omit L, L∗ from these examples, as they will be clear from context. 20 Example 2.2.1 (Bijections). K = {(X, R, S) | R, S ⊆ X & X, R, S are all countably infinite}, K∗ = {(X, R, S, T ) | (X, R, S) ∈ K & T is the graph of a bijection R → S}. In this case, K, K∗ are Gδ . Example 2.2.2 (Ramsey’s Theorem). K = {(X, R, S) | R, S partition [X]2 }, K∗ = {(X, R, S, T ) | (X, R, S) ∈ K & T ⊆ X is infinite and homogeneous for the partition R, S}, where [X]2 is the set of two-element subsets of X. Here K is closed and K∗ is Gδ . Ramsey’s Theorem is exactly the statement that every element of K admits an expansion in K∗ . Example 2.2.3 (Linearizations). K = {(X, P ) | P is a partial order on X}, K∗ = {(X, P, L) | (X, P ) ∈ K & P ⊆ L & L is a linear order on X}. K, K∗ are both closed classes of structures. Example 2.2.4 (Vertex colourings). Fix d ≥ 2, and let K = {(X, E) | (X, E) is a connected graph of max degree ≤ d}, K∗ = {(X, E, S0 , . . . , Sd ) | (X, E) ∈ K & S0 , . . . , Sd is a vertex colouring of (X, E)}. Here K, K∗ are Gδ . Example 2.2.5 (Spanning trees). K = {(X, E) | (X, E) is a connected graph}, K∗ = {(X, E, T ) | (X, E) ∈ K & (X, T ) is a spanning subtree of (X, E)}. K, K∗ are both Gδ . Example 2.2.6 (Z-lines). K = {(X, L) | (X, L) is a linear order without endpoints}, K∗ = {(X, L, Z) | (X, L) ∈ K & Z ⊆ X & (Z, LZ) ∼ = (Z, <)}, where < is the usual order on Z. Here K is Gδ , and K∗ is Borel. 21 Example 2.2.7 (Vizing’s Theorem). Fix d ≥ 2, and let K = {(X, E) | (X, E) is a connected graph of max degree ≤ d}, K∗ = {(X, E, S0 , . . . , Sd ) | (X, E) ∈ K & S0 , . . . , Sd is an edge colouring of (X, E)}. Here K, K∗ are Gδ . Vizing’s Theorem states that every element of K admits an expansion in K∗ . Example 2.2.8 (Matchings). A bipartite graph is said to satisfy Hall’s Condition if |A| ≤ |N (A)| for every finite set of vertices A contained in one part of the graph, where N (A) denotes the set of neighbours of A. We say a graph is locally-finite if every vertex has finitely-many neighbours. Let now K = {(X, E) | (X, E) is a connected, bipartite, locally finite graph satisfying Hall’s Condition}, K∗ = {(X, E, M ) | (X, E) ∈ K & M ⊆ E is a perfect matching}. Here K, K∗ are Borel, and by Hall’s Theorem every element of K admits an expansion in K∗ . 2.2.3 Equivariant and random expansions Let (K, K∗ ) be an expansion problem, X be a countably infinite set, Γ ≤ SX be a subgroup of SX and Z ⊆ K(X) be Γ-invariant. We say a map f : Z → K∗ (X) is Γ-equivariant if it commutes with the Γ action, i.e., γ · f (A) = f (γ · A) for all γ ∈ Γ, A ∈ K(X). We call a function f : Z → K∗ (X) an expansion map if f (A) is an expansion of A for all A ∈ K(X). In this paper, we will always consider the case where Γ is a countably infinite group acting on X = Γ by multiplication on the left. It may also be interesting to consider the more general setting where the action of Γ on X is not free, though we do not explore this here. Let Γ be a countably infinite group. Given a Γ-invariant Borel set Z ⊆ K(Γ), we say Z admits Γ-equivariant expansions to K∗ if there is a Borel Γ-equivariant expansion map Z → K∗ (Γ). If Z = K(Γ), we say K admits Γ-equivariant expansions to K∗ . If K is a Gδ class of structures, we say K admits Γ-equivariant expansions to K∗ generically if Z admits a Γ-equivariant expansion to K∗ for some Γ-invariant dense Gδ set Z ⊆ K(Γ). We let P (K(Γ)) denote the space of probability Borel measures on K(Γ). Note that the action of Γ on K(Γ) gives rise to an action of Γ on P (K(Γ)), where γ · µ = γ∗ µ is 22 the push-forward of µ under γ : K(Γ) → K(Γ). We say µ is Γ-invariant if it is fixed by the Γ-action, in which case we say µ is an invariant random K-structure on Γ. If µ is an invariant random K-structure on Γ, we say K admits Γ-equivariant expansions to K∗ µ-a.e. if Z admits a Γ-equivariant expansion to K∗ for some Γ-invariant µ-conull set Z ⊆ K(Γ). Note that the reduct π : K∗ (Γ) → K(Γ) induces a map π∗ : P (K∗ (Γ)) → P (K(Γ)). If µ (resp. ν) is an invariant random K-structure (resp. K∗ -structure) on Γ, we say ν is a Γ-invariant random expansion of µ to K∗ if π∗ ν = µ. We say µ admits a Γ-invariant random expansion to K∗ if such a ν exists, and that Γ admits random expansions from K to K∗ if this holds for all such µ. Note that if K admits Γ-equivariant expansions to K∗ µ-a.e., then µ admits a Γ-invariant random expansion to K∗ . We may omit Γ from these definitions when it is clear from context. 2.2.4 Countable Borel equivalence relations A countable Borel equivalence relation (CBER) is an equivalence relation E on a standard Borel space X which is Borel as a subset of X 2 , and whose equivalence classes [x]E are countable for all x ∈ X. If Γ is a group acting on a set X, we let EΓX ⊆ X 2 denote the orbit equivalence relation xEΓX y ⇐⇒ ∃γ ∈ Γ(γ · x = y) induced by the action of Γ on X. When Γ is countable, X is standard Borel and the action of Γ is Borel, then EΓX is a CBER. Conversely, by the Feldman–Moore Theorem [FM77], every CBER E on a standard Borel space X is the orbit equivalence relation induced by a Borel action of some countable group Γ on X. By the free part of an action of Γ on X we mean the set F r(X) = {x ∈ X : ∀γ 6= 1Γ (γ · x 6= x)}. Note that when X is standard Borel and the action is Borel, F r(X) is Borel in X. Moreover, if X is Polish and the action of Γ is continuous, then F r(X) is Gδ in X, hence Polish in the subspace topology. Given a CBER E on X and A ⊆ X, we let [A]E = {x ∈ X : ∃y ∈ A(xEy)} denote the (E-)saturation of A. We say A is E-invariant if [A]E = A. Note that by 23 the Feldman–Moore Theorem, if A is Borel then so is [A]E . We call A a complete section for E if X = [A]E . Let E, F be CBER on X, Y respectively. We say that a Borel map f : X → Y is a homomorphism from E to F , denoted f : E →B F , if xEy =⇒ f (x)F f (y). It is a reduction f : E ≤B F if the converse holds as well, i.e., xEy ⇐⇒ f (x)F f (y); an embedding f : E vB F if it is an injective reduction; an isomorphism f : E ∼ =B F cb if it is a surjective embedding; a class-bijective homomorphism f : E →B F if it is a homomorphism for which f : [x]E → [f (x)]F is a bijection for all x ∈ X; and an a invariant embedding f : E viB F if it is a class-bijective reduction. A CBER E is finite if all of its classes are finite, and aperiodic if all of its classes are infinite. Given a CBER E on X, we can always partition X into Borel pieces Y, Z on which E is respectively finite and aperiodic. A CBER E is smooth if E ≤B ∆Y , where Y is equality on a standard Borel space Y . Equivalently, E is smooth iff it admits a Borel selector, i.e., a Borel map s : X → X which is E-invariant (a homomorphism s : E → ∆X ) and so that s(x)Ex for all x ∈ X. The Glimm–Effros Dichotomy for CBER states that for every CBER E, either E is smooth or E0 vB E, where E0 is the eventual equality relation xE0 y ⇐⇒ ∃n∀k(xn+k = yn+k ) on 2N , c.f. [Kec25, Theorem 6.5]. We say E is hyperfinite if E can be written as an increasing union of finite CBER; see [Kec25, Theorem 8.2] for alternate characterizations of hyperfiniteness. In particular, E0 and all smooth CBER are hyperfinite. An invariant probability measure for a Borel action of a countable group Γ on X is a probability Borel measure µ on X such that γ∗ µ = µ for all γ ∈ Γ, where γ∗ µ is the push-forward of µ along γ. An invariant probability measure for a CBER E on X is a probability Borel measure µ on X such that f∗ µ = µ for every Borel bijection f : X → X whose graph is contained in E. If E is the orbit equivalence relation induced by a Borel action of a countable group Γ, then these two notions coincide; see [KM04, Proposition 2.1]. A probability Borel measure µ on X is E-ergodic if µ(A) ∈ {0, 1} for all E-invariant Borel sets A. If Γ is a countable group and X is standard Borel, the shift action of Γ on X Γ is given by (γ · y)(δ) = y(γ −1 δ) for γ, δ ∈ Γ and y ∈ X Γ . If µ is any probability Borel 24 measure on X, then the product measure µΓ is an invariant probability measure on X Γ which concentrates on the free part, i.e., µΓ (F r(X Γ )) = 1. We say E is generically ergodic if A is either meagre or comeagre for all E-invariant Borel sets A. For example, E0 is generically ergodic, as are the orbit equivalence relations induced by the shift actions of countably infinite groups. A CBER E on X is compressible if there is a Borel map f : X → X whose graph is contained in E, and so that f (C) $ C for every E-class C ∈ X/E. By Nadkarni’s Theorem, E is compressible iff it does not admit an invariant probability measure [Nad90; BK96]. 2.2.5 Structures and expansions on CBER Fix a language L and a class K of L-structures. Let E be an aperiodic CBER on a standard Borel space X. A (Borel) L-structure on E is an L-structure A = (X, RA )R∈L with universe X so that (a) RA ⊆ X n is Borel for each n-ary R ∈ L, and (b) each RA only relates elements in the same E-class, i.e., RA (x0 , . . . , xn−1 ) =⇒ x0 E · · · Exn−1 . Given such a structure A and an E-class C ∈ X/E, we let AC denote the restriction of A to C, which is a countable L-structure. We call A a Borel K-structuring of E if AC ∈ K for every C ∈ X/E. Consider now L∗ ⊇ L and a class K∗ of L∗ -structures. If A∗ is an L∗ -structure on E, the reduct A∗ L of A is an L-structure on E. We say that A∗ is an expansion of an L-structure A on E if A∗ L = A. Let A be a Borel K-structuring of E. We say A is Borel expandable to K∗ if it admits an expansion which is a Borel K∗ -structuring of E; we say E is Borel expandable for (K, K∗ ) if this holds for all such A. When E lives on a Polish space X, we say A is generically expandable to K∗ if its restriction to a comeagre invariant Borel set is Borel expandable to K∗ ; we say E is generically expandable for (K, K∗ ) if this holds for all such A. If µ is a probability Borel measure on X, we say A is µ-a.e. expandable to K∗ if its restriction to a µ-conull invariant Borel set is Borel expandable to K∗ ; we say (E, µ) is a.e. expandable for (K, K∗ ) this holds for all such A. 2.3 General results In this section, we assume that all classes of structures are Borel. 25 2.3.1 Universal structurings of CBER We begin by describing an alternate characterization of Borel structures and expansions on CBER which will be useful later; see also [BC24, Definition 3.1]. Let E be an aperiodic CBER on a standard Borel space X. A Borel family of enumerations of E is a Borel map g : X → X N , where N is some countably infinite set, so that gx = g(x) : N → X is a bijection of N with [x]E for all x ∈ X; such maps always exist by the Lusin–Novikov Theorem (c.f. [Kec95, 18.15]). A map ρ : E → G from E to a group G is a cocycle if ρ(y, z)ρ(x, y) = ρ(x, z) for all xEyEz. If g : X → X N is a Borel enumeration of X, there is an associated Borel cocycle ρg : E → SN given by ρg (x, y) = gy−1 ◦ gx . Let L be a language, K be a class of L-structures and g : X → X N be a Borel enumeration of E. Given an L-structure A on E, one gets a map F = FgA : X → ModL (N ) given by setting RF (x) (n1 , . . . , nk ) ⇐⇒ RA (gx (n1 ), . . . , gx (nk )) for k-ary R ∈ L and n1 , . . . , nk ∈ N . Note that if xEy then Rρg (x,y)·F (x) (n1 , . . . , nk ) ⇐⇒ RF (x) (ρg (x, y)−1 (n1 ), . . . , ρg (x, y)−1 (nk )) ⇐⇒ RA (gx (ρg (x, y)−1 (n1 )), . . . , gx (ρg (x, y)−1 (nk ))) ⇐⇒ RA (gy (n1 ), . . . , gy (nk )) ⇐⇒ RF (y) (n1 , . . . , nk ), that is, ρg (x, y) · F (x) = F (y). Call such a map ρg -equivariant. Note that gx : F (x) ∼ = A[x]E for x ∈ X. Conversely, given a ρg -equivariant map F : X → ModL (N ), one can define a Borel L-structure A on E by setting RA (x1 , . . . , xk ) ⇐⇒ RF (x) (gx (x1 )−1 , . . . , gx−1 (xk )) for any k-ary R ∈ L and xEx1 E . . . Exk . It is easy to verify, using the fact that ρg (x, y)F (x) = F (y) for xEy, that this definition does not depend on the choice of x and that gx : F (x) ∼ = A[x]E for x ∈ X. We therefore have that the map A 7→ FgA is a bijective correspondence between L-structures on E and ρg -equivariant maps F : X → ModL (N ). It is easy to see that 26 A is Borel iff FgA is Borel, and that A is a K-structuring of E iff the image of FgA is contained in K(N ), whenever K is a class of L-structures. We remark that if L ⊆ L∗ are languages and A, A∗ are L, L∗ -structures on E, then ∗ A∗ is an expansion of A iff FgA = π ◦ FgA , where π : ModL∗ (N ) → ModL (N ) is the reduct. That is, A admits a Borel expansion iff there is a ρg -equivariant Borel lift F of F A to K∗ (X): K ∗ (X) X F π FA K(X) Fix now a class of L-structures K. In [CK18, Theorem 4.1, Remark 4.3], a universal K-structurable CBER lying over E is constructed, which we denote E ng K. Explicitly, E ng K lives on X × K(N ), and is given by (x, A)(E ng K)(y, B) ⇐⇒ xEy & ρg (x, y) · A = B. The projection π0 : X × K(N ) → X is a class-bijective homomorphism E ng K → E, along which g can be pulled back to a Borel enumeration g̃ of E ng K. If ρg̃ is the associated Borel cocycle, then ρg̃ = ρg ◦ (π0 × π0 ), and it follows that ρg̃ ((x, A), (y, B)) · A = ρg (x, y) · A = B. The canonical K-structure A on E ng K is then the one induced by the projection X × K(N ) → K(N ), which we have observed is ρg̃ -equivariant. Note that while E ng K depends only on the cocycle ρg , A depends on g. The above constructions do not depend on the choice of g, up to canonical Borel isomorphism. That is, given Borel enumerations g : X → X M , h : X → X N of E, let M τ (x) = τg,h (x) = h−1 is Borel and each τ (x) : M → N is a x ◦ gx . Then τ : X → N bijection. Moreover, we have τ (y) ◦ ρg (x, y) = ρh (x, y) ◦ τ (x) for xEy (i.e., ρg , ρh are cohomologous, as witnessed by τ ). In particular, if A is a Borel L-structure on E and FgA , FhA the corresponding maps, then FhA (x) = τ (x) · FgA (x) for x ∈ X. Additionally, the map (x, A) 7→ (x, τ (x) · A) is a Borel isomorphism E ng K ∼ =B E nh K. 27 2.3.2 Expansions on CBER induced by free actions Fix an expansion problem (K, K∗ ) and a countably infinite group Γ. We wish to relate the existence of Γ-equivariant expansions with Borel expansions on CBER induced by free actions of Γ. Suppose E = EΓX is a CBER induced by a free Borel action of Γ on a standard Borel space X. This gives rise to a Borel enumeration g : X → X Γ of E, namely gx (γ) = γ −1 · x. In this case, ρg (x, γx) = γ ∈ SΓ , so ρg : E → Γ. In particular, we X×K(Γ) have that E ng K = EΓ is the orbit equivalence relation induced by the diagonal action of Γ on X × K(Γ). Thus, the characterization of Borel K-structurings of E described in Section 2.3.1 gives: Proposition 2.3.1. Let L be a language, K be a Borel class of L-structures and Γ be a countably infinite group. Fix a free Borel action of Γ on a standard Borel space X. There is a canonical bijection A 7→ F A between the set of Borel K-structurings of EΓX and the set of Γ-equivariant Borel maps X → K(Γ), defined by setting RF (x) (γ1 , . . . , γn ) ⇐⇒ RA (γ1−1 · x, . . . , γn−1 · x) A for x ∈ X, Borel K-structurings A on EΓX , n-ary relation symbols R ∈ L and γ1 , . . . , γn ∈ Γ. Proposition 2.3.2. Let (K, K∗ ) be an expansion problem and Γ be a countably infinite group. Fix a free Borel action of Γ on a standard Borel space X and a Borel Kstructuring A of EΓX , and let f : X → K(Γ) be the corresponding equivariant Borel map. There is a canonical bijection between Borel expansions A∗ of A and equivariant Borel maps g : X → K∗ (Γ) satisfying π ◦ g = f , where π : K∗ (Γ) → K(Γ) is the reduct from L∗ to L. Remark 2.3.3. If X ⊆ F r(K(Γ)) is invariant and Borel then there is a canonical Borel K-structuring of EΓX corresponding to the inclusion X → K(Γ). More generally, if Z is a standard Borel space on which Γ acts and X ⊆ Z × K(Γ) is Borel, invariant and free for the diagonal Γ action on the product, then there is a canonical Borel K-structuring of EΓX corresponding to the projection Z × K(Γ) ⊇ X → K(Γ). In particular, this gives a weak correspondence between equivariant expansions on Γ and expansions of CBER induced by free actions of Γ. 28 Proposition 2.3.4. Let (K, K∗ ) be an expansion problem and Γ be a countably infinite group. (1) If K(Γ) admits a Borel equivariant expansion, then every CBER induced by a free Borel action of Γ is Borel expandable. (2) An invariant Borel set X ⊆ F r(K(Γ)) admits a Borel equivariant expansion iff the canonical K-structuring of EΓX admits a Borel expansion. (3) Suppose there is a free Borel action of Γ on a standard Borel space Z admitting an invariant measure µ, and so that the canonical K-structure A on Z × K(Γ) is λ-a.e. expandable for every Γ-invariant probability Borel measure λ on Z × K(Γ) whose push-forward to Z is µ. Then Γ admits random expansions. Proof. (1) Let h be such an expansion. For any Borel equivariant f : X → K(Γ), h ◦ f : X → K∗ (Γ) is Borel, equivariant, and satisfies π ◦ h ◦ f = f , so this follows by Proposition 2.3.2. (2) This follows immediately from Proposition 2.3.2. (3) For any invariant random K-structure ν on Γ, take λ = ν ×µ. Then λ is Γ-invariant, so there is a λ-conull invariant Borel set X ⊆ K(Γ) × Z and an expansion A∗ of AX. Let f : X → K∗ (Γ) be the corresponding equivariant map, and let κ = f∗ (λ). Then κ is an invariant random K∗ -structure on Γ, and π∗ κ = (π ◦ f )∗ λ = (projK(Γ) )∗ λ = ν. 2.3.3 Uniform random expansions Let (K, K∗ ) be an expansion problem, Γ be a countably infinite group, and µ be an invariant random K-structure on Γ. In general, it is possible that µ admits an invariant random expansion to K∗ , but K does not admit equivariant expansions to K∗ µ-a.e. That is, there may be some invariant random expansion ν of µ to K∗ , but no Borel equivariant expansion map f : K(Γ) ⊇ Z → K∗ (Γ), defined on a µ-conull set Z, so that ν = f∗ µ (see e.g. Remark 2.4.10). On the other hand, we will see that for Examples 2.2.1, 2.2.4, and 2.2.6, every invariant random K-structure on Γ which admits an invariant random expansion to K∗ admits such an expansion of the form f∗ µ, where f : K(Γ) ⊇ Z → K∗ (Γ) is a Borel equivariant expansion map defined on a µ-conull set Z. Moreover, in these cases, we shall see that this holds uniformly in µ: there is a single function f that works for all such µ. That is, there is an invariant Borel set Z ⊆ K(Γ) and Borel equivariant expansion map 29 f : Z → K∗ (Γ) so that µ admits an invariant random expansion to K∗ iff µ(Z) = 1, in which case f∗ µ gives such an expansion. One can view such a function f as both classifying exactly when invariant random expansions exist, as well as giving uniformly all possible invariant random expansions. When such an f exists, one can further characterize exactly when a Borel K-structuring of a CBER is a.e. expandable, for any CBER induced by a free Borel action of Γ: Proposition 2.3.5. Let (K, K∗ ) be an expansion problem and Γ be a countably infinite group. Suppose that there is a Borel Γ-invariant set Z ⊆ K(Γ) which admits a Γ-equivariant expansion to K∗ , and such that the following holds: For every invariant random K-structure µ on Γ, µ admits an invariant random expansion to K∗ iff µ(Z) = 1. Then for any CBER E on X induced by a free Borel action of Γ, any E-invariant probability Borel measure µ on X and any Borel K-structuring A of E, A is µa.e. expandable to K∗ if and only if F A (x) ∈ Z for µ-almost every x ∈ X, where F A : X → K(Γ) is the equivariant map from Proposition 2.3.1. Proof. If F A (x) ∈ Z for µ-almost every x ∈ X, then by Proposition 2.3.2, the composition of F A with the expansion map Z → K∗ (Γ) gives a Borel expansion of A to K∗ on an invariant µ-conull set. Conversely, suppose that Y ⊆ X is an E-invariant Borel µ-conull set and A∗ is a Borel expansion of AY to K∗ . Let π : K∗ (Γ) → K(Γ) denote the reduct, and note that ∗ ∗ F A = π ◦ F A , so that F∗A µ is an invariant random expansion of F∗A µ. By assumption, this implies that F∗A µ(Z) = 1, i.e., F A (x) ∈ Z for µ-a.e. x ∈ X. 2.3.4 Invariant random expansions on CBER We consider now a notion of invariant random structures and random expansions on CBER. When E is a CBER arising from a free Borel action of Γ with an invariant measure, this will correspond to a weakening of the hypotheses of Proposition 2.3.4(3), and we will show in this case that the existence of random expansions on E corresponds exactly to the existence of random expansions on Γ. Crucially, this notion of random expansion will be purely in terms of the CBER with no reference to Γ, allowing us to compare the existence of random expansion between various groups. Let (K, K∗ ) be an expansion problem, fix an aperiodic CBER E on a standard Borel space X and let g : X → X N be a Borel enumeration of E. 30 An invariant random K-structuring of E (with respect to g) is an invariant probability Borel measure for E ng K. If µ is an invariant probability Borel measure for E, an invariant random K-structuring of (E, µ) (with respect to g) is an invariant random K-structuring ν of E whose push-forward along the projection X × K(N ) → X is µ. We say an invariant random K∗ -structuring κ of E is an expansion of an invariant random K-structuring ν of E if the push-forward of κ along the reduct X × K∗ (N ) → X × K(N ) is ν. If h : X → X M is another Borel enumeration of E and τ = τg,h : X → M N the induced Borel map described in Section 2.3.1, we recall that (x, A) 7→ (x, τ (x) · A) is a Borel isomorphism E ng K ∼ =B E nh K. It is easy to see that the following diagram commutes, where π denotes the reduct from K∗ to K. X × K∗ (N ) (idX ,π) (idX ,τ ) X × K∗ (M ) X × K(N ) projX idX (idX ,τ ) (idX ,π) X × K(M ) X projX X If µ is an invariant probability Borel measure on E, we say (E, µ) admits random expansions from K to K∗ if for every invariant random K-structure ν on (E, µ) there is an invariant random K∗ -structure κ on (E, µ) that is an expansion of ν. We say E admits random expansions from K to K∗ if (E, µ) admits invariant random expansions from K to K∗ for every invariant probability Borel measure µ on E. By the prior remarks, these definitions do not depend on the choice of Borel enumeration of E. The following key fact relates invariant random expansions between CBER and groups. Special cases of this have been shown, for example with linearizations in [Alp22]; we note here that it holds more generally for all expansion problems. Proposition 2.3.6. Let (K, K∗ ) be an expansion problem and Γ be a countably infinite group. The following are equivalent: 1. Γ admits random expansions from K to K∗ ; 2. Every CBER E induced by a free Borel action of Γ admits random expansions from K to K∗ ; 3. There is a CBER E induced by a free Borel action of Γ and an E-invariant probability Borel measure µ, such that (E, µ) admits random expansions from K to K∗ . 31 Proof. (1) =⇒ (2): Let E be a CBER on X induced by a free Borel action of Γ. By the remarks at the start of Section 2.3.2, we may assume that E n K is the orbit equivalence relation on X × K(Γ) arising from the diagonal Γ action. If ν is an invariant probability Borel measure on X × K(Γ), then the push-forward of ν along the projection X × K(Γ) → K(Γ) gives an invariant random K-structure ν 0 on Γ. By (1), there is an invariant random K∗ -structure κ on K∗ (Γ) which is an expansion of ν 0 . Let now {κA }A∈K(Γ) be the measure disintegration of κ with respect to ν 0 over the reduct K∗ (Γ) → K(Γ), and {νA × δA }A∈K(Γ) be a measure disintegration of ν with respect to ν 0 over the projection X × K(Γ) → K(Γ) (see for example [Kec95, 17.35]). R Then it is straightforward to check that λ = (νA × κA )dν 0 (A) is a Γ-invariant probability Borel measure on X × K∗ (Γ) which is an extension of ν. (2) =⇒ (3): Consider e.g. the free part of the shift of Γ on 2Γ . (3) =⇒ (1): Suppose (E, µ) admits random expansions form K to K∗ , where E is induced by a free Borel action of Γ on X preserving the probability Borel measure µ. Let ν be an invariant random K-structure on Γ, and consider µ × ν on X × K(Γ). This is Γ-invariant; hence it admits an expansion κ, and the push-forward of κ along the projection X × K∗ (Γ) → K∗ (Γ) is an invariant random expansion of ν. Recall now that two countably infinite groups Γ, Λ are orbit equivalent if there is a CBER E on a standard Borel space X admitting an invariant probability Borel measure and free Borel actions of Γ, Λ on X with E = EΓX = EΛX . A CBER is said to be measure-hyperfinite if, for every invariant probability Borel measure µ, it is hyperfinite when restricted to a Borel invariant µ-conull set (see also [Kec25, Sections 8.5, 9]). For example, all countable infinite amenable groups are orbit equivalent and their actions generate measure-hyperfinite CBER [Dye59; OW80]. Theorem 2.3.7. Let (K, K∗ ) be an expansion problem and Γ, Λ be countably infinite groups. If Γ, Λ are orbit equivalent, then Γ admits random expansions from K to K∗ iff Λ admits random expansions from K to K∗ . In particular, the following are equivalent: 1. Γ admits random expansions from K to K∗ for all amenable groups Γ; 2. Γ admits random expansions from K to K∗ for some amenable group Γ; and 3. every aperiodic measure-hyperfinite CBER admits random expansions from K to K∗ . 32 Proof. Let E be a CBER on a standard Borel space X admitting an invariant probability Borel measure µ, and fix free Borel actions of Γ, Λ on X inducing E. Suppose Γ admits random expansions from K to K∗ . By (1) =⇒ (2) of Proposition 2.3.6 (E, µ) admits random expansions from K to K∗ , and hence by (3) =⇒ (1) of Proposition 2.3.6 so does Λ. By Proposition 2.3.6, it is clear that (1) ⇐⇒ (2) ⇐= (3). If E is an aperiodic measure-hyperfinite CBER and µ is an E-invariant probability Borel measure, then by restricting to a µ-conull set we see that E is hyperfinite, hence generated by a free Z action. Thus (1) =⇒ (3) by Proposition 2.3.6. We note that converse holds as well: Proposition 2.3.8. Let Γ, Λ be countably infinite groups and suppose that for every expansion problem (K, K∗ ), Γ admits random expansions from K to K∗ iff Λ admits random expansions from K to K∗ . Then Γ, Λ are orbit equivalent. Proof. Let K be the class of countable sets (in the empty language), and take K∗ to be a class of structures so that K∗ -structurability of a CBER E corresponds exactly to E arising from a free Borel action of Γ (see e.g. [CK18, Section 3.1]). We claim that Γ admits an invariant random K∗ -structure. To see this, fix a free Borel action of Γ on a standard Borel space X preserving a probability Borel measure µ (e.g. the free part of the shift on 2Γ ). By definition, EΓX admits a Borel K∗ -structure A∗ . By Proposition 2.3.1 this corresponds to a Γ-equivariant Borel map F A : X → K∗ (Γ), so F∗A µ is an invariant random K∗ -structure on Γ. As K(Γ) is a singleton, Γ admits invariant random expansions from K to K∗ . Thus, by our assumption, Λ admits random expansions from K to K∗ . Fix a free Borel action of Λ on a standard Borel space Y preserving a probability Borel measure ν. By (1) =⇒ (2) of Proposition 2.3.6 there is an (E n K∗ )-invariant probability Borel measure κ on Y × K∗ (Λ). Note that E n K∗ admits a Borel K∗ -structuring, and therefore arises from a free Borel action of Γ. Thus the actions of Γ, Λ on Y × K∗ (Λ) witness the orbit equivalence of Γ, Λ. Remark 2.3.9. This is really an observation about invariant random structures on groups: Two countably infinite groups Γ, Λ are orbit equivalent if and only if, for every class K of structures, there is an invariant random K-structure on Γ exactly when there is an invariant random K-structure on Λ. 33 Finally, we remark that random expansions always exist for closed classes of structures on amenable groups. Proposition 2.3.10. Let (K, K∗ ) be an expansion problem. If K∗ is closed, then Γ admits random expansions from K to K∗ for every countably infinite amenable group Γ. Proof. Let µ be an invariant random K-structure on Γ, and let π : K∗ (Γ) → K(Γ) denote the reduct. Then π∗ : P (K∗ (Γ)) → P (K(Γ)), where P (X) denotes the space of probability Borel measures on a space X. Note that π∗ is continuous and P (K∗ (Γ)) is compact metrizable [Kec95, 17.22, 17.28], so in particular A = (π∗ )−1 (µ) ⊆ P (K∗ (Γ)) is compact metrizable. By the invariance of µ and the equivariance of π, we see that A is Γ-invariant. It is also easy to see that it is non-empty: as π is continuous every fibre is compact, so we may choose in a Borel way some νx ∈ P (π −1 (x)) for x ∈ K(Γ) R and let ν = νx dµ ∈ A (c.f. [Kec95, 28.8]). As Γ is amenable there is a fixed point in A, which is an invariant random expansion of µ. 2.3.5 Generic equivariant expansions If K is Gδ classes of structures of L-structures and Φ is an isomorphism-invariant property of L-structures, we say the generic element of K satisfies Φ if for some countably infinite set X, KΦ (X) = {A ∈ K(X) : Φ(A)} is comeagre in K(X). We note that this does not depend on the choice of X: If f : X → Y is a bijection, this induces a homeomorphism K(X) → K(Y ) taking KΦ (X) to KΦ (Y ). In this section, we show that if K is a Gδ class of structures and the generic element of K has trivial algebraic closure, then the question of whether K admits Γ-equivariant expansions to K∗ generically does not depend on the group Γ. Definition 2.3.11. Let L be a language and A be a countable L-structure with universe X. We say A has the weak duplication property (WDP) if, for every A0 ∈ AgeX (A) and any finite F ⊆ X, there is an embedding of A0 into A whose image is disjoint from F . We say that A is definable from equality if for all n-ary R ∈ L and n-tuples x̄, ȳ in X with the same equality type, RA (x̄) ⇐⇒ RA (ȳ). (The equality type of x̄ is the set of pairs (i, j) with x̄i = x̄j .) For F ⊆ X, let AutF (A) denote the group of automorphisms of A that fix F pointwise, i.e., such that f (x) = x for x ∈ F . We say A has trivial algebraic closure (TAC) 34 if, for every finite F ⊆ X, the action of AutF (A) on X \ F has no finite orbits. See e.g. [Hod93, Section 4.2] or [CK18, 50] for alternative characterizations. Example 2.3.12. Let K be the class described in one of Examples 2.2.1 to 2.2.3, 2.2.5, and 2.2.6. The generic element of K has TAC and is not definable from equality. On the other hand, if K is the class of connected graphs of maximum degree d (c.f. Examples 2.2.4 and 2.2.7) then no element of K has TAC: If G ∈ K and (u, v) is an edge of G, then the orbit of v under the action of Aut{u} (G) has size at most d. Theorem 2.3.13. Let (K, K∗ ) be an expansion problem. Suppose that K is Gδ and the generic element of K has TAC and is not definable from equality. Then the following are equivalent: 1. For every countably infinite group Γ, K admits Γ-equivariant expansions to K∗ generically. 2. There exists a countably infinite group Γ for which K admits Γ-equivariant expansions to K∗ generically. In particular, this applies to Examples 2.2.1 to 2.2.3, 2.2.5, and 2.2.6 by Example 2.3.12. Lemma 2.3.14. Let A be an L-structure on a countably infinite set X. Then A has TAC if and only if, for every A0 ∈ AgeX (A) with universe F , every F0 ⊆ F , every embedding f : F0 → X of A0 F0 into A and every finite G ⊆ X there is an embedding g of A0 into A which extends f and such that g(F \ F0 ) ∩ G = ∅. Proof. ( =⇒ ) Let A0 , F, F0 , G be as in the lemma and suppose that A has TAC. By Fraïssé’s Theorem and [Hod93, Theorem 7.1.8] A is homogeneous, so we may assume wlog that f is the identity. By Neumann’s Separation Lemma, there is some g ∈ AutF0 (A) such that g(F \ F0 ) ∩ G = ∅, in which case we may take gF . ( ⇐= ) Note first that A is homogeneous, i.e., every isomorphism between finite substructures of A extends to an automorphism of A. To see this, by [Hod93, Lemma 7.1.4] it suffices to show that if A0 ∈ AgeX (A) has universe F , F0 ⊆ F and f : F0 → X is an embedding of A0 F0 into A, then f can be extended to an embedding of A0 into A, which follows from our assumption (taking G = ∅). Let now F be a finite set and x ∈ X \ F in order to show that x has infinite orbit under AutF (A). Let C ⊆ X be finite and let A0 = A(F ∪ {x}). By assumption, there is an embedding f of A0 into A that is the identity on F and such that f (x) ∈ / C. By 35 homogeneity, this can be extended to an automorphism of A, so that in particular the orbit of x is not contained in C. As C, x were arbitrary, we conclude that A has TAC. In particular, if A has TAC then A is homogeneous and has the WDP (to see the latter, take F0 = ∅ and G = F in the lemma). Remark 2.3.15. If K is a class of structures and X a countably infinite set, the set of structures in K(X) which have the WDP (resp. are definable from equality, have TAC) is Gδ (resp. closed, Gδ ) in K(X). Lemma 2.3.16. Let K be a class of structures with the WDP that are not definable from equality, and Γ be a countably infinite group. For any A ∈ K(Γ) there is some B ∈ F r(K(Γ)) isomorphic to A. If K is Gδ , then F r(K(Γ)) is a dense Gδ set in K(Γ). Proof. Let A ∈ K(Γ) and fix an enumeration {γn }n∈N of Γ. Let R ∈ L be a relation such that RA is not definable from equality. We will construct an increasing sequence of finite partial bijections fn : Γ → Γ so that γn is in the domain of f2n and in the range of f2n+1 , and moreover so that for all n there is a tuple x̄ such that x̄, γn x̄ are contained in the domain of f2n and RA (f2n (x̄) ⇐⇒ ¬RA (f2n (γn x̄)). Assuming this S has been done, we let f = n fn and B = f −1 (A). Then B ∼ = A ∈ K(Γ), and for all γ ∈ Γ there is some x̄ such that RB (x̄) ⇐⇒ RA (f (x̄)) ⇐⇒ ¬RA (f (γ x̄)) ⇐⇒ ¬RB (γ x̄), so that γB 6= B for all γ ∈ Γ. We construct these maps as follows. Set f−1 = ∅ for convenience. Given f2n , we let f2n+1 be an arbitrary extension with γn in its range. Suppose now we have constructed f2n−1 : F → G. Because A has the WDP and RA is not definable from equality, there are tuples ȳ, z̄ with the same equality type so that RA (ȳ) ⇐⇒ ¬RA (z̄) and the sets ȳ, z̄, G are pairwise disjoint. Since Γ is infinite, we can find a tuple x̄ with the same equality type as ȳ so that x̄, γn x̄, F are pairwise disjoint. We then define f2n to extend f2n−1 by sending x̄ 7→ ȳ, γn x̄ 7→ z̄ and, if f2n (γn ) has not already been defined, setting it to any element of Γ not already in the range of f2n . If K is Gδ , then F r(K(Γ)) is a Gδ set as the action of Γ is continuous. Moreover, if A0 ∈ AgeΓ (K) has universe F and A ∈ N (A0 ), then the same construction starting 36 instead with the f−1 as the identity on F gives B ∈ N (A0 ) ∩ F r(K(Γ)), so the free part is dense. Lemma 2.3.17. Let K be a Gδ class of structures with TAC and Γ be a countably infinite group. The set of A ∈ K(Γ) with the following extension property is a dense Gδ set in K(Γ): (∗) Let A0 , A1 ∈ AgeΓ (K) be L0 -structures with disjoint universes F, G respectively. Let F0 ⊆ F and f : (F \ F0 ) ∪ G → Γ be an injection. If A0 F0 v A and A0 , A1 embed into A, then there is some γ ∈ Γ such that the map F0 3 x 7→ x, (F \ F0 ) ∪ G 3 x 7→ γf (x) embeds both A0 , A1 into A. Proof. Fix A0 , A1 , F, G, F0 , f as in (∗) and let a : F → Γ, b : G → Γ be injections. Let U be the set of all A ∈ K(Γ) such that, if A0 F0 v A and a, b are embeddings of A0 , A1 into A, then the conclusion of (∗) holds for A. We will show that U is open and dense. As there are only countably many choices for A0 , A1 , F, G, F0 , f, a, b, the intersection of all such U is a dense Gδ set whose elements satisfy (∗). It is clear that U is open. To see that it is dense, fix B0 ∈ AgeΓ (K) and let A ∈ N (B0 ). If A0 F0 * A or some a, b is not an embedding of A0 , A1 into A then A ∈ U . Otherwise, let B0 have universe H, and assume wlog that F0 ⊆ H. Because A has TAC and by Lemma 2.3.14, there is an embedding g0 : F → Γ of A0 into A extending the identity on F0 so that g0 (F \ F0 ) ∩ H = ∅. By the WDP, there is an embedding g1 : A1 → A whose image is disjoint from H ∪ g0 (F ). Fix γ so that H ∩ γf ((F \ F0 ) ∪ G) = ∅, and let h : Γ → Γ be a bijection so that the following hold: h is the identity on H, h(γf (x)) = g0 (x) for x ∈ F \ F0 and h(γf (x)) = g1 (x) for x ∈ G. Let B = h−1 (A). Then B ∈ N (B0 ), B ∼ = A and B satisfies (∗) as witnessed by γ, so B ∈ U . Proof of Theorem 2.3.13. Clearly (1) =⇒ (2). (2) =⇒ (1): As the class K0 of elements of K with TAC that are not definable from equality is a comeagre Gδ set in K, we may assume wlog that K = K0 . Let Γ, Λ be countably infinite groups and Z ⊆ K(Λ) be a Borel comeagre Λ-invariant set that admits a Λ-equivariant expansion to K∗ . Let fn ∈ SΛ be a dense sequence of bijections, i.e., a sequence such that for every finite partial bijection Λ → Λ there is some fn extending it. Since SΛ acts on K(Λ) 37 by homeomorphisms, we can assume wlog that Z is Gδ and that fn (A) ∈ Z for all A ∈ Z, n ∈ N. By Lemma 2.3.16, we may also assume that Z ⊆ F r(K(Λ)). Let X = F r(K(Γ)), which by Lemma 2.3.16 is a dense Gδ set in K(Γ), and let E = EΓX . Let A be the canonical K-structuring of E. We will show that there is a Borel expansion of A restricted to a comeagre Γ-invariant set Y ⊆ X, and hence by Proposition 2.3.4(2) there is a Borel Γ-equivariant expansion map Y → K∗ (Γ). As Γ was arbitrary, this proves (1). Our proof strategy is as follows: We find a Borel Γ-invariant comeagre set Y ⊆ X and a free Λ-action on Y so that EΛY = EΓY . By Proposition 2.3.1 applied to AY , this gives a Λ-equivariant Borel map F : Y → K(Λ). We will ensure that the image of F is contained in Z. By Proposition 2.3.2, this implies the existence of a Borel expansion of AY , completing the proof. Below, we let ∀∗ denote the category quantifier: If W is a topological space and A ⊆ W is Baire-measurable, ∀∗ wA(x) means that A is comeagre (see [Kec95, 8.J]). Let G denote the intersection graph of E. That is, the vertices of G are finite subsets of X which are contained in a single E-class, and aGb ⇐⇒ a 6= b & (a ∩ b 6= ∅). By the proof of [KM04, Lemma 7.3], we may fix a countable Borel colouring c of G. Let (Rn , γn , δ̄n ) be a sequence of triples so that (1) the tuples in {δ̄n , γn δ̄n : n ∈ N} are pairwise disjoint, (2) for every n, Rn ∈ L, and if Rn has arity k then δ̄ ∈ Γk , and (3) for every R ∈ L of arity k and every equality type of tuples of length k, there are infinitely many n with Rn = R such that δ̄n has this equality type. Let On = {A ∈ X : RnA (δ̄n ) & ¬RnA (γn δ̄n )} and Bn+1 = On \ S i<n Oi , n ∈ N. We also set B0 = X \ S n On . Claim 2.3.18. Suppose A ∈ K(Γ) satisfies (∗) from Lemma 2.3.17. Then |Γ·A∩Bn | = ∞ for infinitely many n. Proof. Let N ∈ N be arbitrary. We may find some n ≥ N so that RnA is not definable from equality, and there are tuples x̄, ȳ with the same equality type as δ̄n so that RnA (x̄) & ¬RnA (ȳ). By the WDP, we may assume x̄, ȳ are disjoint. For i < n, find x̄i with the same equality type as δ̄i which are disjoint from each other and from x̄, ȳ. By (∗), there is some γ so that RnA (γ δ̄n ), ¬RnA (γγn δ̄n ) and for i < n, if RiA (x̄i ) then RiA (γγi δ̄i ) and otherwise ¬RA (γ δ̄i ). It follows that γ −1 A ∈ Bn . / 38 For α ∈ NN , we define an equivalence relation E α on X as follows: We set E0α to be α α α ∪ [y]E α ) = α(n), equality. Given Enα , we set xEn+1 y if either xEnα y or x E n y, c([x]En n S S α α and x, y ∈ i<n Bi . We then set E = n En . Note that if [x]Enα is not a singleton for some α, n, then x ∈ i<n Bi . We note also that we may analogously construct Eis for s ∈ N<N , i ≤ |s|, and that Eiα = Eis for all s such s, i and α ⊇ s. We set E s = E|s| . S Claim 2.3.19. ∀x ∈ X∀∗ α([x]E = [x]E α ). Proof. Fix x ∈ X and let yEx. We show that the set of α with y ∈ [x]E α is open and dense. It is clearly open, as if yE α x then yEnα x for some n. To see it is dense, S fix s ∈ Nn . We may assume wlog that x, y ∈ i<n Bi . But then any α ⊇ s with α(n) = c([x]E s ∪ [y]E s ) satisfies yE α x. As there are only countably many yEx, the set of all α for which [x]E = [x]E α is dense Gδ . / Let now L0 = (fλ )λ∈Λ , where each fλ is a binary relation, and let Λ = (Λ, fλΛ )λ∈Λ be the L0 -structure where fλΛ (δ) = λδ is interpreted as (the graph of) multiplication on the left by λ. Let α, β ∈ NN . We define an L0 -structure Λα,β on X as follows. We will define an S increasing sequence of structures Λα,β on Enα and then take Λα,β = n Λα,β n n . We will ensure at every stage n of this process that, if C is an Enα -class, then Λα,β n C will be isomorphic to a substructure of Λ. Fix a Borel linear order < on X and an enumeration of Λ. We define Λα,β by setting 0 Λα,β 0 fid (x) = x for all x ∈ X, and leaving the other relations undefined. Suppose now α,β α that we have defined Λα,β n , in order to define Λn+1 . Let C be an En+1 -class. If C is an α,β Enα -class, then we define Λα,β on C. Otherwise, C is the union n+1 to be equal to Λn α of two En -classes C0 , C1 . Order them so that the <-least element of C0 is <-below the <-least element of C1 . For each Ci , as Λα,β n Ci is isomorphic to a substructure of Λ, there is a unique embedding fi : Ci → Fi ⊆ Λ taking the <-least element of Ci to the identity in Λ. We then take λ to be the β(n)-th element of Λ, in our fixed enumeration, satisfying F0 ∩ F1 · λ = ∅. We define now Λα,β n+1 C to be the pullback of Λ(F0 ∪ F1 · λ) via the injection (f0 ∪ f1 · λ) : (C0 ∪ C1 ) → (F0 ∪ F1 · λ), where f1 · λ denotes the map x 7→ f1 (x) · λ. (Note that we are multiplying F1 , f1 by λ on the right. This is because Λ is defined in terms of multiplication on the left, and this commutes with multiplication on the right.) 39 n s,t As with the E α , we may define Λs,t = Λs,t n . k for s, t ∈ N and k ≤ n, and we let Λ αn,βn Note that Λα,β for all α, β. n = Λ Claim 2.3.20. ∀∗ x ∈ X∀∗ α, β(Λα,β [x]E α ∼ = Λ). Proof. Fix x ∈ X satisfying (∗) of Lemma 2.3.17. Note that by construction, Λα,β [x]E α embeds into Λ for all α, β, and hence there is a unique embedding f α,β : Λα,β [x]E α → Λ taking x to the identity. Thus it suffices to show that the set of all α, β for which f α,β is surjective is a dense Gδ set. To see this, fix λ ∈ Λ. We will show that the set of all α, β so that λ ∈ f α,β ([x]E α ) is a dense open set. As there are only countably many such λ, this completes the proof. Note that we may define similarly f s,t : [x]E s → Λ for s, t ∈ Nn . Then f α,β = S αn,βn , so it is clear that the set of α, β with λ in its image is an open set. To see nf that it is dense, fix s, t ∈ Nn and consider f s,t . If λ is in the image of f s,t then any α, β extending s, t satisfies that λ is in the image of f α,β . So suppose otherwise. By Claim 2.3.18, there is some yEx so that [y]E s is a singleton. It is easy to see that if u is an extension of s whose new values are all c([x]E s ∪ {y}), then for sufficiently long u we have yE u x. Pick such a u of minimal length, so that at stage |u| of the construction s of E u we merge [x]E with {y}. Let C0 , C1 denote these two sets, ordered as in the construction of Λα,β , and let fi : Ci → Fi ⊆ Λ be the corresponding embeddings. If v is any extension of t of length |u|, m = v(|u| − 1) and λm is the m-th element of Λ such that F0 ∩ F1 · λm = ∅, then Λu,v C ∼ = Λ(F0 ∪ F1 · λm ) via the map f = f0 ∪ f1 · λm . In particular, if x ∈ C0 and f0 (x) = δ, then f u,v = f · δ −1 . On the other hand, if −1 x ∈ C1 and f1 (x) = δ, then f u,v = f · λ−1 m δ . We will show that we can choose m so that λ = f u,v (y), and hence such that λ is in the image of f α,β for all α, β extending u, v. As s, t were arbitrary, the set of all such α, β is dense and we are done. Consider now two cases. If x ∈ C0 and f0 (x) = δ, then by assumption λδ ∈ / F0 . Since C1 is a singleton, F1 contains only the identity. Pick m so that λm = λδ. For such an m we have f u,v (y) = f1 (y)λm δ −1 = λδδ −1 = λ. On the other hand, if x ∈ C1 and f1 (x) = δ, then by assumption λδ ∈ / F1 . In this case again F0 contains only the identity. Pick m with λm = δ −1 λ−1 . For such an m we have −1 f u,v (y) = f0 (y)λ−1 = λδδ −1 = λ. / m δ For all α, β, let X α,β ⊆ X denote the set of all x for which [x]E α = [x]E and Λα,β [x]E ∼ = Λ. There is a free Borel action of Λ on this set, namely the action where 40 λ · x = fλΛ (x). By Proposition 2.3.1, the structure A on EX α,β = EΛX to a Λ-equivariant Borel map F α,β : X α,β → K(Λ). α,β α,β gives rise Claim 2.3.21. ∀∗ x ∈ X∀∗ α, β(x ∈ X α,β =⇒ F α,β (x) ∈ Z). Proof. Fix any A ∈ X that is isomorphic to an element of Z and satisfies (∗) of Lemma 2.3.17. Note that the set of all such A is comeagre (for the first condition, note that any bijection Λ → Γ gives a homeomorphism K(Λ) → K(Γ) and consider the image of Z). If A ∈ X α,β , let B α,β = F α,β (A). We will show that the set of all α, β for which A ∈ X α,β =⇒ B α,β ∈ Z is a dense Gδ set. Fix s, t ∈ Nn and let C s,t = [A]E s . Let f s,t : C s,t → I s,t ⊆ Λ be the unique embedding of Λs,t C s,t into Λ which takes A to the identity (note that this f s,t is the same as the one described in the proof of the previous claim). Let also Ds,t = {γ : γ −1 A ∈ C s,t }, let g s,t : Ds,t → C s,t be the map γ 7→ γ −1 A and let hs,t = f s,t ◦g s,t . Let B s,t = hs,t (ADs,t ). It is easy to see, by Proposition 2.3.1, that B s,t v B α,β whenever α ⊇ s, β ⊇ t and A ∈ X α,β . Let now Un be a sequence of dense open sets in K(Λ) so that Z = n Un . We will show that for all N , the set of α, β so that A ∈ X α,β =⇒ B α,β ∈ UN is dense and open. Since UN is open, T A ∈ X α,β =⇒ [B α,β ∈ UN ⇐⇒ ∃L0 ⊆ L∃n(N (B αn,βn L0 ) ⊆ UN )]. Thus the set of all such α, β is exactly the set of α, β satisfying ∃L0 ⊆ L∃n(N (B αn,βn L0 ) ⊆ UN ), which is clearly open, so it remains to show that it is dense. That is, we need to show that for all s, t, there are u, v extending s, t so that N (B u,v L0 ) ⊆ UN for some finite L0 ⊆ L. Fix s, t ∈ Nn . By assumption, there is some A0 ∈ Z that is isomorphic to A. As Z is closed under the functions fn described at the start of the proof, we may assume that B s,t v A0 . Since A0 ∈ Z ⊆ UN , there is some L0 -structure B0 ∈ AgeΛ (K) so that A0 ∈ N (B0 ) ⊆ UN ∩ N (B s,t L0 ). Let F be the universe of B0 , so that B0 = (A0 L0 )F and wlog F ⊇ I s,t . We will show that there are u, v extending s, t so that B u,v L0 = B0 , which would complete the proof. We will show how to do this assuming that F = I s,t t {λ}. By repeating this argument recursively we can handle all finite F . 41 By the proof of Claim 2.3.18, there is some m > n, a finite structure A0 ∈ AgeΓ (A) and an injection f from the universe of A0 to Γ \ {1Γ } so that if x 7→ γf (x) is an embedding of A0 into A, then γ −1 A ∈ Bm . It is clear from the proof that we can also assume that the universe of A0 is disjoint from Ds,t . Now B0 embeds into A, as it embeds into A0 ∼ = A, so by (∗) there is some γ so that (a) the map x 7→ γf (x) is an embedding of A0 into A and (b) the map (hs,t )−1 ∪ {(λ, γ)} is an embedding F → Γ of B0 into A. By (a) and our choice of A0 , γ −1 A ∈ Bm so the E s -class of γ −1 A is a singleton. Extend s to a sequence u of length m + 1 by setting the new values to be c([A]E s ∪ {γ −1 A}). −1 u = [A]E s and [A]E u = [A]E s ∪ {γ Thus [A]Em A}. By the proof of the previous claim, there is an extension v of t of length m + 1 so that f u,v (γ −1 A) = λ. We claim now that B u,v L0 = B0 . To see this, note that Du,v = Ds,t ∪ {γ}, I u,v = I s,t ∪ {λ} = F and hu,v = (hs,t ∪ {(γ, λ)}). Now B u,v = hu,v (ADu,v ), so this follows immediately from (b). / By the Kuratowski–Ulam Theorem [Kec95, 8.41] and the claims above, we may fix some α, β so that X α,β is comeagre and x ∈ X α,β =⇒ F α,β (x) ∈ Z for the generic x ∈ X. In particular, there is a comeagre Γ-invariant Borel set Y ⊆ X α,β such that F α,β (Y ) ⊆ Z, which proves (1) by the remarks at the start of the proof. 2.3.6 Enforcing smoothness Let (K, K∗ ) be an expansion problem. We are broadly interested in relating the class of Borel expandable CBER with natural classes of CBER such as being smooth or compressible. In this section, we give some sufficient conditions for an expansion problem (K, K∗ ) to be Borel expandable for exactly the class of smooth CBER. Proposition 2.3.22. Let (K, K∗ ) be an expansion problem. If there is a Borel expansion map f : K(N) → K∗ (N), then every smooth aperiodic CBER is Borel expandable for (K, K∗ ). Note that we do not require f to satisfy any additional properties (such as equivariance). Proof. Let E be a smooth aperiodic CBER on X. Since E is smooth, we can identify every E-class with N in a Borel way, i.e., there is a Borel enumeration g : X → X N so that if xEy then g(x) = g(y) (for example take any Borel enumeration h of E and a selector s for E and let g = h ◦ s). If F : X → K(N) is Borel and E-invariant, 42 then composing this with f gives a Borel E-invariant map F ∗ = f ◦ F : X → K∗ (N) so that F ∗ (x) is an expansion of F (x) for all x. By the correspondence described in Section 2.3.1, it follows that E is Borel expandable for (K, K∗ ). Remark 2.3.23. In many cases of interest (including all of the examples in Section 2.2.2), a Borel expansion map K(N) → K∗ (N) can easily be shown to exist, for example by recursively constructing an expansion for a given A ∈ K(N), or via an application of the Compactness Theorem (see e.g. [Kec95, 28.8]). Note that such constructions depend crucially on the given enumeration of the universe of A, and hence are not in general equivariant. Definition 2.3.24. Let (K, K∗ ) be an expansion problem and E be a class of aperiodic CBER. We say (K, K∗ ) enforces E if an aperiodic CBER E belongs to E whenever E is K-structurable and Borel expandable for (K, K∗ ). When E is the class of aperiodic smooth CBER we say that such (K, K∗ ) enforces smoothness. Proposition 2.3.25. Let (K, K∗ ) be an expansion problem. If some hyperfinite, compressible, aperiodic CBER is not Borel expandable for (K, K∗ ), then (K, K∗ ) enforces smoothness. In particular, this holds if some aperiodic CBER is not generically expandable for (K, K∗ ). Proof. Let E be a hyperfinite, compressible, aperiodic CBER and let F be any nonsmooth aperiodic CBER. By the Glimm-Effros Dichotomy and compressibility we have E viB F (c.f. [DJK94]). It is now easy to see that if F is Borel expandable then so is E, hence if E is not Borel expandable then neither is F . The second part follows from the first by [KM04, Theorem 12.1, Corollary 13.3]. Corollary 2.3.26. Let (K, K∗ ) be an expansion problem. Suppose K is Gδ and there is a countably infinite group Γ with F r(K(Γ)) 6= ∅ so that there is no Borel equivariant expansion map X → K∗ (Γ) for any comeagre invariant Borel set X ⊆ F r(K(Γ)). Then (K, K∗ ) enforces smoothness. Proof. Let A be the canonical Borel K-structuring of E = EΓ . By our assumption on Γ and Proposition 2.3.4(2), A does not admit a Borel expansion when restricted to any comeagre E-invariant Borel set, and in particular E is not generically expandable. The conclusion follows by Proposition 2.3.25. F r(K(Γ)) 43 Remark 2.3.27. By Theorem 2.3.13, if the generic element of K has TAC and not definable from equality then the hypotheses of Corollary 2.3.26 hold for some group Γ iff they hold for all groups Γ. We note the following weak converse: Proposition 2.3.28. Let (K, K∗ ) be an expansion problem. If some non-smooth CBER admits a Borel K-structuring and K admits a Borel Γ-equivariant expansion to K∗ for some countably infinite group Γ then (K, K∗ ) does not enforce smoothness. Proof. Let E be a hyperfinite compressible CBER. If some non-smooth CBER admits a Borel K-structuring, then so does E (as E invariantly embeds into any non-smooth CBER). Now consider the orbit equivalence relation E of the shift of Γ on 2Γ . This action is generically ergodic, hence by [KM04, Theorem 12.1, Theorem 13.3] it is hyperfinite, compressible and non-smooth on an invariant dense Gδ set Y ⊆ F r(2Γ ). Thus EΓY admits a Borel K-structuring, and by Proposition 2.3.4(1) it is Borel expandable, so (K, K∗ ) does not enforce smoothness. 2.4 Examples In this section, we will consider in detail definable expansion problems for Examples 2.2.1 to 2.2.8. We summarize what is known for these problems in Table 2.1. 44 Expansions on CBER Bijections Ramsey’s Theorem Linearizations Vertex colouring Spanning trees Z-lines Vizing’s Theorem Matchings Borel expandable Generically expandable Expandable a.e. Smooth1 Smooth [GX24] Smooth All1 ?3 Smooth Smooth5 Smooth [CJMST20] All ? (CE) ? (CE) All1 All ? (CE) ? (PP6) ? (CE7, PP8) Classified ? (CE) ? (CE) All1 ?3 Classified All [GP20] ? (CE7, PP8) Equivariant expansions on groups Borel expansions Bijections None1 Ramsey’s Theorem None Linearizations None Vertex colouring All1,2 Spanning trees ?4 Z-lines None Vizing’s Theorem None Matchings None [CJMST20] Generic expansions Expansions a.e. Random expansions All None None All1 All None ? (PP6) All9 Classified ? (CEΓ ) ? (CEΓ ) Classified1 ?4 Classified All2 [GP20] ? (CEΓ 7, PP8) Classified ? (CEΓ ) Amen. [Alp22] All1 ?4 Classified All [GP20] ? (CEΓ 7, PP8) Classified: In the sense of Section 2.3.3 and Proposition 2.3.5. CE: There are counterexamples coming from free continuous actions of Γ, for all Γ. CEΓ There are counterexamples for all countably infinite groups Γ. PP: There are partial positive results (see the corresponding section for details and references). 1 Essentially [KST99]. 2 On the free part F r(K(Γ)). 3 This lies somewhere between hyperfinite and treeable. 4 This lies somewhere between amenable and treeable. 5 Smooth for d ≥ 3 [CJMST20], All for d = 2 [KST99]. 6 Subexponential growth [BD25] and bipartite [BW23]. 7 See [Lac88; CK13; Kun24; BKS22]. 8 See [LN11; MU16; CM17; BKS22; BCW24; BPZ24; KL23]. 9 All for graphs of bounded degree d > 2 (None for d = 2). Table 2.1: A summary of known results for Examples 2.2.1 to 2.2.8 45 2.4.1 Bijections Fix K, K∗ as in Example 2.2.1, that is, K = {(X, R, S) | R, S ⊆ X & X, R, S are all countably infinite}, K∗ = {(X, R, S, T ) | (X, R, S) ∈ K & T is the graph of a bijection R → S}. Theorem 2.4.1 (Essentially [KST99]). (K, K∗ ) enforces smoothness, and (E, µ) is not a.e. expandable for any CBER E and any E-invariant probability Borel measure µ. Proof. Let E be a non-smooth aperiodic CBER. By the Glimm–Effros Dichotomy, E0 vB E. By [KST99, Theorem 1.1] E0 is not Borel expandable, and it follows that E is not Borel expandable either. If µ is an E-invariant probability Borel measure, then (E, µ) is not a.e. expandable by the same argument as in [KST99, Section 1] for the shift on 2Z . More generally, if µ is an ergodic invariant probability Borel measure for a CBER E on a standard Borel space X and A, B ⊆ X are Borel, then µ(A) = µ(B) iff A = (X, A, B) admits a Borel expansion µ-a.e. (see e.g. [KM04, Lemma 7.10]). A similar proof gives a characterization of the invariant random K-structures that admit invariant random expansions. Theorem 2.4.2. Let Γ be a countably infinite group. There is a Borel Γ-invariant set X ⊆ K(Γ) and a Borel equivariant expansion map f : X → K∗ (Γ) such that, for all invariant random K-structures µ on Γ, µ admits a random expansion to K∗ if and only if µ(X) = 1, in which case f∗ µ gives such an expansion. Moreover, let A = (Γ, A, B) ∼ µ for an invariant random K-structure µ on Γ. If µ admits an invariant random expansion then P[1Γ ∈ A] = P[1Γ ∈ B], and the converse holds when µ is ergodic. In particular, if E is a CBER on Z induced by a free Borel action of Γ, µ is an E-invariant probability Borel measure and A is a Borel K-structuring of E, then A is µ-a.e. expandable to K∗ iff F A (z) ∈ X for µ-a.e. z ∈ Z. Proof. The “in particular” part follows immediately from Proposition 2.3.5. Let Γ = {γn } be an enumeration of Γ. For A, B ⊆ Γ, define sets XnA,B recursively by ! XnA,B = A \ [ m<n A,B Xm ∩ B\ ! [ m<n A,B Xm · γm · γn−1 . 46 The collections {XnA,B }, {XnA,B ·γn } consist of pairwise disjoint sets, and the map taking S S γ ∈ XnA,B to φA,B (γ) = γ · γn is a bijection from n XnA,B ⊆ A to n XnA,B · γn ⊆ B. It is easy to see by induction that Xnγ·A,γ·B = γ · XnA,B for γ ∈ Γ, so the map (A, B) 7→ φA,B is Γ-equivariant. Additionally, either dom(φA,B ) = A or ran(φA,B ) = B: If γ ∈ A\dom(φA,B ), γ 0 ∈ B\ran(φA,B ) and γn = γ −1 γ 0 then γ ∈ XnA,B , a contradiction. We let X ⊆ K(Γ) be the set of all A = (Γ, A, B) such that φA,B is a bijection A → B. It is clear that X is invariant and Borel, and that A 7→ (A, φA,B ) defines a Borel equivariant expansion f : X → K∗ (Γ). Now let µ be an invariant random K-structure on Γ. If µ(X) = 1 then f∗ µ is an invariant random expansion of µ. If ν is an invariant random expansion of µ and (Γ, A, B, T ) ∼ ν then P[1Γ ∈ A] = P[∃γ((1Γ , γ) ∈ T )] = X P[(1Γ , γ) ∈ T ] = γ X P[(γ −1 , 1Γ ) ∈ T ] γ (2.1) = P[∃γ((γ, 1Γ ) ∈ T )] = P[1Γ ∈ B], and since ν is a random expansion of µ we have that P[1Γ ∈ A] = P[1Γ ∈ B] for (Γ, A, B) ∼ µ. Suppose now µ is ergodic and P[1Γ ∈ A] = P[1Γ ∈ B] for (Γ, A, B) ∼ µ. As in (2.1), we find that P[1Γ ∈ dom(φA,B )] = P[1Γ ∈ ran(φA,B )]. If P[A = dom(φA,B )] = 1 then P[1Γ ∈ B] = P[1Γ ∈ A] = P[1Γ ∈ dom(φA,B )] = P[1Γ ∈ ran(φA,B )], and it follows that P[B = ran(φA,B )] = 1. Similarly, if P[B = ran(φA,B )] = 1 then P[A = dom(φA,B )] = 1. By ergodicity, one of these must hold, and so P[A = dom(φA,B )] = P[B = ran(φA,B )] = 1 and hence µ(X) = 1. It remains to show that if µ admits an invariant random expansion then µ(X) = 1, and for this it suffices to prove that if ν is an invariant random K∗ -structure on Γ and (Γ, A, B, T ) ∼ ν then P[(A, B) ∈ X] = 1. By considering an ergodic decomposition of K∗ (Γ) (cf. [Kec25, Theorem 5.12]) we may assume ν is ergodic, in which case this follows by the same argument as in the previous paragraph. It is not hard to verify that the set X constructed in the proof of Theorem 2.4.2 is a F r(K(Γ)) dense Gδ set in K(Γ), so that the canonical K-structuring of EΓ admits a Borel expansion on a comeagre invariant Borel set. More generally, we have the following: 47 Theorem 2.4.3. Every aperiodic CBER is generically expandable. In particular, K admits Γ-equivariant expansions to K∗ generically for every countably infinite group Γ. Proof. The second part follows from the first by Proposition 2.3.4. In order to prove the first part, let E be an aperiodic CBER on a Polish space X and let R, S ⊆ X be Borel sets that have infinite intersection with every E-class. Let xF y ⇐⇒ xEy & (x ∈ R ⇐⇒ y ∈ R) & (x ∈ S ⇐⇒ y ∈ S). By (the proof of) [KM04, Corollary 13.3], there is a comeagre E-invariant Borel set for which the aperiodic part of F C is compressible. As the finite part A of F C is smooth, and hence so is E[A]E , it suffices to prove the following: Claim 2.4.4. Suppose F is compressible. Then there is a Borel bijection R → S whose graph is contained in E. Proof. We note first that we may assume that R \ S, S \ R have infinite intersection with every E-class. Indeed, E is smooth on the set of points for which this is false, and one can easily construct expansions on smooth CBER. By taking our bijection to be the identity on R ∩ S, we may therefore assume that R, S are disjoint. Fix a Borel action of a countable group Γ on X so that xEy ⇐⇒ ∃γ ∈ Γ(γx = y). Fix an enumeration (γn )n∈N of Γ and for x ∈ R let n(x) be the least n such that γn x ∈ S. Let f (x) = (γn(x) , n(x)), so that f : R → S × N is a Borel embedding of F R into F S × IN , where IN is the equivalence relation on N with a single equivalence class. Since F S is compressible, there is a Borel isomorphism g : F S × IN → F S such that xF g(x, n) for all x ∈ S, n ∈ N (see e.g. the proof of [DJK94, Proposition 2.5]). Thus g ◦ f : R → S is a Borel injection whose graph is contained in E. Since F R is compressible, the proof of [DJK94, Proposition 2.3] applied to g ◦ f gives a Borel bijection R → S whose graph is also contained in E. / 2.4.2 Ramsey’s Theorem Fix K, K∗ as in Example 2.2.2, that is, K = {(X, R, S) | R, S partition [X]2 }, K∗ = {(X, R, S, T ) | (X, R, S) ∈ K & T ⊆ X is infinite and homogeneous for the partition R, S}. 48 Theorem 2.4.5. Let Γ be a countably infinite group. Then K does not admit Γ-equivariant expansions to K∗ generically. In particular, (K, K∗ ) enforces smoothness. Proof. The proof of the second part follows from the first by Corollary 2.3.26. Suppose now by way of contradiction that f : X → K∗ (Γ) was a Borel equivariant expansion on an invariant dense Gδ set X ⊆ K(Γ). Since an expansion f (x) in this case is just a choice of a subset of Γ that is homogeneous for the partition given by x ∈ X, we may view f as an equivariant Borel map X → 2Γ . Note that the generic element of K has TAC and is not definable from equality, so by Lemma 2.3.17 we may assume that for any A0 ∈ AgeΓ (K) and any A ∈ X there is some γ ∈ Γ such that γA0 v A. By shrinking X further still we may assume that f is continuous. Now fix an arbitrary A ∈ X and let T = f (A) ⊆ Γ. Fix γ0 6= γ1 ∈ T . By continuity there is some A0 ∈ AgeΓ (K) so that A0 v A and γ0 , γ1 ∈ f (B) whenever B ∈ X ∩ N (A0 ). By equivariance, γγ0 , γγ1 ∈ f (B) whenever B ∈ X ∩ N (γA0 ). Let now F be the universe of A0 , and assume wlog that γ0 , γ1 ∈ F . We consider the case where [T ]2 ⊆ RA ; the case where [T ]2 ⊆ S A is handled identically. Fix γ ∈ Γ so that F ∩ γF = ∅, and let A1 = (F ∪ γF, RA1 , S A1 ) ∈ AgeΓ (K) satisfy A0 v A1 , γA0 v A1 and {γ0 , γγ0 } ∈ S A1 . Let δ be such that δA1 v A. Then δA0 v A, δγA0 v A so δγ0 , δγ1 , δγγ0 ∈ T . Also, {δγ0 , δγ1 } ∈ RA and {δγ0 , δγγ0 } ∈ S A . This contradicts the fact that T is homogeneous for the partition (RA , S A ). Remark 2.4.6. In [GX24], it is shown that an aperiodic CBER is Borel expandable for (K, K∗ ) iff it is smooth, and in particular that (K, K∗ ) enforces smoothness. Theorem 2.4.5, along with Proposition 2.3.22 and Remark 2.3.23, give an alternative proof of this result. Gao and Xiao consider more generally the case of k-colourings of sets of size n (with the appropriate modifications made to K, K∗ ) for k, n ≥ 2 [GX24, Theorem 1.3]; we note that our proof of Theorem 2.4.5 holds in this more general setting as well (where one takes in this case A1 to contain n copies of A0 ). In [GX24, Section 3], a variation of the Ramsey extension property is introduced that enforces (and is actually equivalent to) hyperfiniteness. This extension property involves choosing in an “almost invariant” way (see [GX24, Definition 3.1]) an expansion on each E-class, and in particular does not fit into our framework of expansion problems. 49 Example 2.4.7. We showed above that there is a Borel K-structuring of a CBER X which does not admit an expansion on any comeagre set. Our proof gave a K-structure (X, R, S) such that, when viewing (X, R) as a graph, each connected component is isomorphic to the Rado graph. Below are two more such examples. In the first, (X, R) is acyclic, and in the second it is bipartite. 1. By [KST99, Proposition 6.2] there is a Borel acyclic graph G0 ⊆ E0 on 2N for which every Borel independent set is meagre. Let A = (2N , G0 , E0 \ G0 ), and note that this is a Borel K-structuring of E0 . If X ⊆ 2N is Borel and E0 -invariant and A∗ = (AX, T ) is a Borel expansion of AX, then T ∩ C is G0 -independent for every E0 -class C ⊆ X (as G0 is acyclic), so T is meagre. Since E0 is generated by the action of a countable group of automorphisms of 2N , X = [T ]E0 is meagre as well. In particular, X is not comeagre, and hence E0 is not generically expandable. 2. (Kechris) Consider an irrational rotation R on the circle T and let E = ERT . Define f : E \ ∆T → 2 by f (x, y) = 1 iff Rn (x) = y for some even n ∈ Z, where ∆T ⊆ T2 denotes the diagonal. Let A = (T, f −1 (0), f −1 (1)) and note that A is a Borel K-structuring of E. If X ⊆ T is Borel and E-invariant and A∗ = (AX, T ) is a Borel expansion of AX. then clearly f takes the value 1 on T . One can easily extend T to a Borel set T ⊆ A ⊆ X so that R2 (A) = A. It follows, as R2 is generically ergodic, that A, and hence A ∪ R(A) = X, are meagre. In particular, X is not comeagre, and hence E is not generically expandable. By Proposition 2.3.25, these examples give alternative proofs that (K, K∗ ) enforces smoothness. Additionally, both of these examples are not a.e. expandable for the Haar measure, by a similar ergodicity argument. Proposition 2.4.8 provides another example that is not expandable a.e. Proposition 2.4.8. Let Γ be a countably infinite group and let µ be the law of the partition (R, S) of [Γ]2 obtained by including every pair {γ, δ} in R independently with probability p, 0 < p < 1. Then µ is an invariant random K-structure on Γ that does not admit an invariant random expansion. 50 Proof. It is clear that µ is Γ-invariant. Suppose there was an invariant random expansion ν of µ, and let (R, S, T ) ∼ ν. We may view R as a graph on Γ, which is almost surely isomorphic to the Rado graph, and view T as either an infinite clique or an infinite independent set. For any finite F ⊆ Γ, RF is the random graph on F whose edges are included independently with probability p. The expected size of the largest clique in RF is Θ(log(|F |)), and hence so is the expected size of the largest independent set [Bol01, Theorem 11.4]. Thus E[|F ∩ T |] ∈ O(log(|F |)). On the other hand, E[|F ∩ T |] = E[ X γ∈F 1γ∈T ] = X P[γ ∈ T ] = |F | · P[1Γ ∈ T ] γ∈F by invariance of ν. Taking |F | → ∞ we find that P[1Γ ∈ T ] = 0, and hence that T = ∅ almost surely, a contradiction. 2.4.3 Linearizations Fix K = {(X, P ) | P is a partial order on X}, K∗ = {(X, P, L) | (X, P ) ∈ K & P ⊆ L & L is a linear order on X}, as in Example 2.2.3. The expansion problem for invariant random K-structures on groups has been studied in [GLM24; Alp22]. In particular, Alpeev has shown that: Theorem 2.4.9 ([Alp22, Corollary 1.1]). A countable group Γ admits random expansions from K to K∗ if and only if Γ is amenable. Remark 2.4.10. Note that even in the case when Γ is amenable and all invariant random partial orders admit invariant random extensions to linear orders, we cannot expect to find an equivariant map f : K(Γ) → K∗ (Γ) for which an extension is given by the pushfoward measure along f . This is unlike the case of bijections, or (as we will see below) for vertex colourings or Z-lines. As a trivial example, note that the empty partial order is a fixed point in K(Γ), and there is no equivariant expansion of this partial order when Γ is not left-orderable. We give a more interesting example in Theorem 2.4.11. Theorem 2.4.11. Let E be an aperiodic CBER. There is a Borel K-structuring of E that is not µ-a.e. expandable for any E-invariant probability Borel measure µ. 51 Proof. We may assume that E is not smooth, as no smooth CBER admits an invariant probability Borel measure. It suffices to show this for the CBER F = ∆2N × E0 . Indeed, suppose A were a Borel K-structuring of F with this property. By [Kec25, Corollary 8.14], we can assume that E lives on 2N × 2N and that F ⊆ E. Then every E-invariant measure is also F -invariant, A is also a Borel K-structuring of E, and if A∗ is an expansion of A on E (restricted to some invariant Borel set) then its restriction to each F -class is an expansion of A on F . It follows immediately that A witnesses that this holds for E as well. Let F0 be the index 2 subequivalence relation of E0 given by xF0 y ⇐⇒ xE0 y & |{i : x(i) 6= y(i)}| = 0 mod 2. We define a Borel K∗ -structuring L of F0 as follows: For xF0 y, we say xLy if either P x = y or i<n x(i) = 0 mod 2, where n is maximal with x(n) 6= y(n). This is clearly reflexive and anti-symmetric. To see that it is transitive, let xLyLz and let l, m, n be maximal such that x(l) 6= y(l), y(m) 6= z(m) and x(n) 6= z(n). It is clear that l 6= m. P P If l < m, then m = n and i<m x(i) = i<m y(i) = 0 mod 2, so xLz. Otherwise P l > m, so l = n and i<l x(i) = 0 mod 2, and so xLz. Note that in particular, for all xLz there are only finitely many y with xLyLz. Indeed, if n is maximal with x(n) 6= z(n), then y must agree with either x or z at all coordinates m > n. Additionally, it is clear that the restriction of L to every F0 -class is total. Below, we identify ik for i ∈ 2, k ≤ ∞ with the constant sequence of length k with value i, and let _ denote concatenation of sequences. (We also abuse notation and write i for i1 .) We view L as a Borel K-structuring of E0 . Suppose that X ⊆ 2N is Borel and E0 invariant, and that L0 is a Borel expansion of LX to K∗ . We claim that µ(X) = 0, where µ is the Haar measure on 2N . To see this, define f : 2N \ {1_ 0∞ } → 2N \ {0∞ } by f (0_ i_ x) = 1_ (1 − i)_ x, f (1_ 0n_ 1_ i_ x) = 0n+1_ 1_ (1 − i)_ x. It is clear that this is a Borel function whose graph is contained in F0 . Moreover, it is not hard to verify that f (x) is the immediate successor of x in L, whenever this is defined. It follows in particular that L orders every F0 -class with order-type Z, except for [0∞ ]F0 , [1_ 0∞ ]F0 , and that f generates L (i.e., xLy ⇐⇒ ∃nf n (x) = y). 52 L0 C x0 y0 LC0 g LC1 g(y0 ) g(x0 ) Figure 2.1: There is some LC0 -maximal x0 for which x0 L0 g(x0 ). Let g : 2N → 2N be the map which flips the first coordinate of every sequence. Then g is a Borel involution, its graph is contained in E0 , and x F0 g(x) for all x. It is also ∼ easy to verify that g is an isomorphism F0 = F0 , and that it is order-reversing for L. We will use L0 to find a Borel F0 -invariant set Y ⊆ X so that every E0 X-class contains exactly one F0 -class in Y . It is easy to see that µ is F0 -invariant and F0 -ergodic, so this implies that Y is either µ-null or µ-conull. It clearly cannot be µ-conull, so µ(Y ) = 0 and hence µ(X) = 0 as well. Let now C be an E0 -class in X. We show how to choose an F0 -class from C in a uniformly Borel way, and then take Y to be the set of all such choices. If 0∞ ∈ C, then we choose [0∞ ]F0 , so suppose this is not the case. Let C0 , C1 be the two F0 -classes in C. If some Ci is an initial segment of L0 C, then we choose Ci . Otherwise, we claim that for i ∈ 2 there is a unique pair (xi , yi ) ∈ Ci so that xi Lyi , xi L0 g(xi ) and g(yi )L0 yi (see Fig. 2.1). We then take x to be the lexicographically least element of {x0 , x1 , y0 , y1 } and choose [x]F0 . We show this for i = 0, the case i = 1 being symmetric. As C1 is not an initial segment of L0 C, there are x ∈ C0 , y ∈ C1 with xL0 y. If yLg(x), then xL0 g(x). Otherwise, g(x)Ly so g(y)Lx (as g is order-reversing) and so g(y)L0 y. Thus, by possibly setting x = g(y), we may assume that xL0 g(x). Clearly there is an L-maximal such x, as we have assumed that C0 is not an initial segment of L0 C. We may then take x0 = x, y0 = f (x). Let now A be the K-structuring of ∆2N ×E0 given by pulling back L along the projection proj2 : 2N × 2N → 2N to the second coordinate (note that this is a class-bijective map ∆2N × E0 →cb B E0 ). We claim that for any (∆2N × E0 )-invariant probability Borel 53 δA0 δγA0 δγ0 δγγ0 δγ1 δγγ1 Figure 2.2: A copy of δA1 in A. The solid arrows are relations in A1 , and the dashed arrows are relations that are forced to exist in L. measure µ and any invariant Borel set X ⊆ 2N × 2N , if there is a Borel expansion of AX to K∗ then µ(X) = 0. By considering an ergodic decomposition we may assume that µ is ergodic, in which case it is equal to the Haar measure on {x} × 2N for some x ∈ 2N , so this follows by the analogous fact for L on E0 . With respect to category, we have the following: Theorem 2.4.12. Let Γ be a countably infinite group. Then K does not admit Γ-equivariant expansions to K∗ generically. In particular, (K, K∗ ) enforces smoothness. Proof. The second part follows from the first by Corollary 2.3.26. Suppose by way of contradiction that there was such an expansion f : X → K∗ (Γ). By shrinking X, we may assume that X is Gδ , f is continuous, and by Lemma 2.3.17 (and the fact that there is a unique generic partial order) that for every A0 ∈ AgeΓ (K) and every A ∈ X there is some γ ∈ Γ with γA0 v A. Fix an arbitrary A = (Γ, P ) ∈ X and let (Γ, L) = f (A), so that P ⊆ L. Since P contains a copy of every element of AgeΓ (K) we may in particular find γ0 , γ1 ∈ Γ that are P -incomparable. Suppose wlog that γ0 L γ1 . By continuity and equivariance, there is some A0 ∈ AgeΓ (K) so that A0 v A and for all γ ∈ Γ and B ∈ X, if γA0 v B then γγ0 is less than γγ1 in f (B). Let now F be the universe of A0 and assume wlog that γ0 , γ1 ∈ F . Find some γ ∈ Γ so that F ∩ γF = ∅, and let A1 = (F ∪ γF, P1 ) ∈ AgeΓ (K) be a structure with universe F ∪ γF so that A0 v A1 , γA0 v A1 and such that γ1 P1 γγ0 and γγ1 P1 γ0 . Let δ be such that δA1 v A. Then δA0 v A, δγA0 v A so δγ0 L δγ1 and δγγ0 L δγγ1 . On the other hand, as δP1 ⊆ P ⊆ L, we have δγ1 L δγγ0 and δγγ1 L δγ0 (see Fig. 2.2.) It follows that δγ0 L δγ1 L δγ0 , a contradiction. 54 y x I = Ix = Iy Ln+1 {z ∈ C \ Zn : I = Iz } ) L̄n (C ∩ Zn ) Figure 2.3: Extending L̄n to L̄n+1 . We consider now expansions on CBER. Proposition 2.4.13. Let E be a CBER on a standard Borel space X and let P be a Borel partial order on E. Then the set of Borel subsets Y ⊆ X for which P Y admits a Borel linearization on EY forms a σ-ideal. Proof. This class is clearly closed under taking Borel subsets. Suppose now that S Y = n Yn and for every n there is a Borel linearization Ln of P Yn on EYn . Let S Zn = i<n Yn . We will recursively construct an increasing sequence L̄n of Borel S linearizations of P Zn on EZn and then take L = n L̄n . We begin by setting L̄0 = ∅. Suppose now that we have constructed L̄n , and let C be an EZn+1 -class in order to define L̄n+1 C. For x ∈ C \ Zn , let Ix = {y ∈ C ∩ Zn : ∃z ∈ C ∩ Zn (z P x & y L̄n z)}. Note that Ix is an initial segment of L̄n (C ∩ Zn ). Now for x, y ∈ C, we say x L̄n+1 y iff one of the following hold (c.f. Fig. 2.3): 1. x, y ∈ Zn and x L̄n y, 2. x ∈ Zn , y ∈ / Zn and x ∈ Iy , 3. x ∈ / Zn , y ∈ Zn and y ∈ / Ix , 4. x, y ∈ / Zn and Ix $ Iy , or 5. x, y ∈ / Zn , Ix = Iy and x Ln+1 y. One easily checks that L̄n+1 is a linear order on Zn+1 . 55 In particular, if a partial order P on a CBER E can be decomposed into a countable union of chains and antichains, then P admits a Borel linearization on E. Proposition 2.4.14. Let T be a Borel locally countable directed tree on a standard Borel space X whose (undirected) connected components form a CBER E, and let P be the smallest partial order containing T . Then (E, P ) admits a Borel linearization. Proof. For all xEy, define d(x, y) as follows: Consider the unique (undirected) path from x to y in T . Weigh each edge in this path by 1 if it occurs in T , and by −1 if its reverse appears in T , and let d(x, y) be the sum of the edge weights along this path. Fix a Borel linear order ≤ on X and define x L y if either d(x, y) > 0, or d(x, y) = 0 and x ≤ y. It is straightforward to verify that L is a linear order on E extending P (for transitivity, note that d(x, z) = d(x, y) + d(y, z) whenever these are defined). 2.4.4 Vertex colourings In this section, we fix d ≥ 2 and let K = {(X, E) | (X, E) is a connected graph of max degree ≤ d}, K∗ = {(X, E, S0 , . . . , Sd ) | (X, E) ∈ K & S0 , . . . , Sd is a vertex colouring of (X, E)}, as in Example 2.2.4. It was shown in [KST99, Proposition 4.6] that every CBER is Borel expandable for (K, K∗ ), and therefore that (K, K∗ ) admits random expansions by Proposition 2.3.4. Their proof essentially establishes the following (see also [BC24, Proposition 4.29]). Theorem 2.4.15 (Essentially [KST99]). Let Γ be a countably infinite group. There is an invariant dense Gδ set F r(K(Γ)) ⊆ X ⊆ K(Γ) which admits a Borel equivariant expansion, and such that X is maximal with this property: For any invariant X % Y , there is no equivariant expansion Y → K∗ (Γ). Interpreted in the language of [BC24], X consists of exactly the set of orbits which satisfy an appropriate separation axiom (as used for example in the proof of [BC24, Proposition 4.29]). Proof. We say G ∈ K(Γ) is bad if there are γ, δ ∈ Γ so that δ G γδ and γG = γ, and G is good otherwise. If G is bad, then so is every graph in its orbit Γ · G, and in this case there is no equivariant expansion map Γ · G → K∗ (Γ). Indeed, suppose c were such an expansion 56 map, and view c(γG) as a map Γ → d + 1 that is a colouring of γG for γ ∈ Γ. Fix some γ, δ so that δ G γδ and γG = G. Then c(G)(γδ) = c(γG)(γδ) = (γc(G))(γδ) = c(G)(δ) by equivariance of c, a contradiction. We take X to be the set of good graphs. Clearly X is a Gδ set containing the free part of K(Γ). It is also not hard to see that it is dense (for example, the free part is dense by Lemma 2.3.16 as the generic element of K satisfies the WDP and is not definable from equality). It thus remains only to show that X admits a Borel equivariant expansion. To see this, it suffices to construct a Borel map f : X → d + 1 so that f (G) 6= f (γ −1 G) whenever G ∈ X, γ ∈ Γ and 1Γ G γ. Indeed, thinking of f (G) as the colour of 1Γ in G, this extends uniquely to an equivariant map g : X → (d + 1)Γ sending G to the colouring g(G)(γ) = f (γ −1 G). It is clear that g is equivariant and Borel, and g(G) is a proper colouring of G because if γ G γδ then 1Γ γ −1 G δ so g(G)(γ) = f (γ −1 G) 6= f (δ −1 γ −1 G) = g(G)(γδ). Let Hn be an enumeration of AgeΓ (K) and let Xn be the set of all G ∈ X ∩ N (Hn ) such that γ −1 G ∈ / N (Hn ) for all neighbours γ of 1Γ in G. It is clear that at most S −1 one of G, γ G ∈ Xn whenever G ∈ X and 1Γ G γ. We claim that X = n Xn . Indeed, as G ∈ X is good and has bounded degree, there is some finite F ⊆ Γ so that GF 6= γ −1 GF for all neighbours γ of 1Γ in G, in which case G ∈ Xn for n such that Hn = GF . Let Yn = Xn \ i<n Xi . The sets Yn partition X and if G ∈ Yn and 1Γ G γ then γ −1 G ∈ / Yn . We now define f : X → d + 1 recursively on each Yn as follows: supposing S f has already been defined on i<n Yi , we define f (G) for G ∈ Yn to be the least element of d + 1 which is not equal to f (γ −1 G) for all neighbours γ of 1Γ in G. S Note that if we replace K with the class of d-regular graphs, then the sets Xn in the previous proof are clopen, so the construction yields a continuous equivariant expansion X → K∗ (Γ). One can also consider vertex colourings with fewer colours. This has been studied extensively in the case of expansions on CBER; we refer the reader to [KM20, Part I] for a survey of this topic. 57 2.4.5 Spanning trees In this section, consider K = {(X, E) | (X, E) is a connected graph}, K∗ = {(X, E, T ) | (X, E) ∈ K & (X, T ) is a spanning subtree of (X, E)}, as in Example 2.2.5. We say a CBER E is treeable if there is a Borel K0 -structuring of E, where K0 is the class of connected trees. Let T denote the class of treeable CBER, and say an expansion problem enforces treeability if it enforces T . A countably infinite group Γ is antitreeable if for every free Borel action of Γ on a standard Borel space X admitting an invariant probability Borel measure, the CBER EΓX is not treeable (c.f. [Kec25, 115]). Theorem 2.4.16. Let E be a CBER. If E is hyperfinite then it is Borel expandable for (K, K∗ ), and if it is Borel expandable for (K, K∗ ) then it is treeable. In particular, 1. (K, K∗ ) enforces treeability; 2. K admits Γ-equivariant expansions to K∗ generically for all countably infinite groups Γ; 3. Γ admits random expansions from K to K∗ for all amenable groups Γ; and 4. Γ does not admit random expansions from K to K∗ for all antitreeable groups Γ. Proof. It is clear that if E is Borel expandable for (K, K∗ ) then it is treeable (consider the complete graph on each E-class), and in particular that (1) holds. To see that hyperfinite CBER are Borel expandable for K, K∗ , let E be a hyperfinite S CBER and A be a Borel K-structuring of E. Write E = n En for an increasing union of finite CBER. We recursively construct an increasing sequence of Borel sets A∗n so that A∗n ⊆ En and for every En -class C, A∗n C is a spanning subforest of AC. We describe the construction of A∗n+1 , given A∗n . Let C be an En+1 -class, G = AC, T = A∗n C. Then T ⊆ G is a forest of trees, so we can easily find a spanning forest T ⊆ T 0 ⊆ G. We set A∗n+1 C = T 0 . As every En+1 -class is finite, it is clear that this can be done in a uniformly Borel way. 58 It follows that F r(K(Z)) admits an equivariant random expansion to K∗ , and by Theorem 2.3.13 this holds for all countably infinite groups Γ. (3) follows by Proposition 2.3.4(3), as every CBER generated by a Borel action of an amenable group is measure-hyperfinite, and (4) is an immediate consequence of Proposition 2.3.6. Thus the class of CBER that are Borel expandable for (K, K∗ ) lies somewhere between the hyperfinite and the treeable CBER. Problem 2.4.17. Is every treeable CBER expandable for (K, K∗ )? Does (K, K∗ ) enforce hyperfiniteness? 2.4.6 Z-lines Let K = {(X, L) | (X, L) is a linear order without endpoints}, K∗ = {(X, L, Z) | (X, L) ∈ K & Z ⊆ X & (Z, LZ) ∼ = (Z, <)}, as in Example 2.2.6. Theorem 2.4.18. Let Γ be a countably infinite group. Then K does not admit Γ-equivariant expansions to K∗ generically. In particular, (K, K∗ ) enforces smoothness. Proof. The second part follows from the first and Corollary 2.3.26. For A0 ∈ AgeΓ (K) and A ∈ K(Γ), let C(A0 , A) = {γ : γA0 v A}. We say A contains A0 densely often if AC(A0 , A) is a dense linear order with at least two points. We claim that the generic element of K(Γ) contains every A0 ∈ AgeΓ (K) densely often. As there are only countably many such A0 , it suffices to show this for some A0 . It is easy to see that the set of all A for which C(A0 , A) contains at least two points is open and dense, so we show that the set of A for which AC(A0 , A) is dense is a dense Gδ set. To see this, we show that for any fixed γ0 , γ1 ∈ Γ, the set of A satisfying γ0 , γ1 ∈ C(A0 , A) & γ0 LA γ1 =⇒ ∃δ(δ ∈ C(A0 , A) & γ0 LA δ LA γ1 ) is dense and open. This set is clearly open. To see that it is dense, fix B0 ∈ AgeΓ (K) and let A ∈ N (B0 ). If γi ∈ / C(A0 , A) for some i ∈ 2 or γ1 LA γ0 , we are done. 59 Otherwise, we may assume that γ0 , γ1 are in the universe of B0 and that γi A0 v B0 for i ∈ 2. Let F be the universe of B0 and fix δ so that F ∩ δγ0−1 F = ∅. Let B1 be a linear order with universe F ∪ δγ0−1 F so that B0 v B1 , δγ0−1 B0 v B1 and γ0 LB1 δ LB1 γ1 . Then for any B ∈ N (B1 ) ⊆ N (B0 ) we have γ0 A0 , γ1 A0 , δA0 v B and γ0 LB δ LB γ1 . Suppose now that there is a comeagre Borel equivariant set X ⊆ K(Γ) and a Borel equivariant expansion f : X → K∗ (Γ). By shrinking X, we may assume that it is Gδ and that f is continuous. We view f as a function X → 2Γ taking A ∈ X to a subset of Γ so that Af (A) ∼ = Z. Fix now some A ∈ X in which every element of AgeΓ (K) appears densely often and let γ0 ∈ f (A) be arbitrary. By continuity and equivariance, there is some A0 ∈ AgeΓ (K) so that A0 v A and whenever γA0 v B ∈ X we have γγ0 ∈ f (B). In particular, C(A0 , A) ⊆ f (A). But A0 appears densely often in A, contradicting the fact that Af (A) ∼ = Z. We consider now the measurable case. For this, we will need the following lemma, due to Lyons and Schramm [LS99], on the existence of “densities” of infinite random subsets of a group Γ (see also [HP24, Section 4.2]). Let Γ be a countably infinite group and let (Zn )n∈N be a random walk on Γ with symmetric step distribution µ whose support generates Γ. (Note that we do not assume µ to be finitely supported.) For γ ∈ Γ, let Pγ denote the law of the random walk (Zn )n∈N starting at γ. Define Ω(Γ, µ) to be the set of all W ⊆ Γ for which there exists r ∈ [0, 1] so that X 1 n−1 lim 1(Zi ∈ W ) = r, Pγ -a.s. for all γ ∈ Γ, n→∞ n i=0 and for W ∈ Ω(Γ, µ) we let Freqµ (W ) be the unique such r. We note that Ω(Γ, µ) ⊆ 2Γ , Freqµ : Ω(Γ, µ) → [0, 1] are Borel and Γ-invariant. (Here, 1(Zn ∈ W ) is equal to 1 when Zn ∈ W and 0 otherwise.) Lemma 2.4.19 (Existence of frequencies [LS99, Lemma 4.2]; c.f. [HP24, Lemma 4.4]). Let Γ be a countably infinite group and µ be a symmetric probability measure on Γ whose support generates Γ. Let ν be an invariant random equivalence relation on Γ and let E ∼ ν. Then ν-almost surely, every equivalence class of E is contained in Ω(Γ, µ). 60 We include a proof of Lemma 2.4.19 in Appendix 2.A, for the reader’s convenience. Theorem 2.4.20. Let Γ be a countably infinite group. There is a Borel Γ-invariant set X ⊆ K(Γ) and a Borel equivariant expansion map f : X → K∗ (Γ) such that, for all invariant random K-structures µ on Γ, µ admits a random expansion to K∗ if and only if µ(X) = 1, in which case f∗ µ gives such an expansion. Moreover, we can choose f so that for all A ∈ X, f (A) picks out an interval I in A with AI ∼ = Z. In particular, if E is a CBER on Z induced by a free Borel action of Γ, µ is an E-invariant probability Borel measure and A is a Borel K-structuring of E, then A is µ-a.e. expandable to K∗ iff F A (z) ∈ X for µ-a.e. z ∈ Z. Proof. The “in particular” part follows immediately from Proposition 2.3.5. For notational convenience, we identify A ∈ K(Γ) with LA . Let κ be a fixed symmetric probability measure on Γ whose support generates Γ. For a given L ∈ K(Γ), let ZL denote the set of all intervals I in L for which LI ∼ = Z. We define X ⊆ K(Γ) to be the set of all L for which supI∈ZL Freqκ (I) exists, is non-zero and is attained by finitely many I ∈ ZL . For such L, we define f (L) to be the L-least interval I ∈ ZL maximizing Freqκ (I). It is clear that f gives a Borel equivariant expansion X → K∗ (Γ). Suppose now that µ is an invariant random K-structure on Γ admitting a random expansion ν. Let (L, S) ∼ ν be a random variable with law ν. We claim that ν-almost surely, for all x, y ∈ S, there are finitely many points between x and y in L. To see this, define g(x, y, L, S), for x, y ∈ Γ, by setting g(x, y, L, S) = 1 if y is the L-least P element of S with x L y, and 0 otherwise. Note that y g(x, y, L, S) is equal to 1 if x P lies between two elements of S, and 0 otherwise. On the other hand, x g(x, y, L, S) is 0 when y ∈ / S, and when y ∈ S it is equal to the size of the interval (z, y] in L, where z is the LS-predecessor of y. Let now G(x, y) = E[g(x, y, L, S)]. Then by the mass transport principle (which in this case follows simply from the invariance of ν), X x G(x, y) = X x G(y, x) = E[ X g(y, x, L, S)] ≤ 1 x for all y ∈ Γ. It follows that the size of the interval [y, z] in L is almost surely finite for all y, z ∈ S. In particular, if we take I(L, S) to be the smallest interval in L containing 61 S, then I(L, S) ∈ ZL almost surely. Thus, by replacing ν with the law of (L, I(L, S)), we may assume that S ∈ ZL . We now show that L ∈ X almost surely, i.e., µ(X) = 1. By Lemma 2.4.19 we may assume that I ∈ Ω(Γ, κ) for all I ∈ ZL , i.e., that Freqκ (I) is defined for all such intervals. By considering an ergodic decomposition of ν (cf. [Kec25, Theorem 5.12]), we may assume that ν is ergodic. Let P[1Γ ∈ S] = r > 0. We claim that Freqκ (S) = r almost surely. Indeed, by ergodicity Freqκ (S) is constant a.s., and by the Dominated Convergence Theorem and invariance we have X X 1 n−1 1 n−1 P[Zi ∈ S] = lim r = r. n→∞ n n→∞ n i=0 i=0 E[Freqκ (S)] = lim It follows that supI∈ZL Freqκ (I) > 0, and by Fatou’s Lemma X [ Freqκ (I) ≤ Freqκ ( ZL ) ≤ 1, I∈ZL so the max is attained by finitely many I ∈ ZL . 2.4.7 Vizing’s Theorem Fix d ≥ 2 and let K = {(X, E) | (X, E) is a connected graph of max degree ≤ d}, K∗ = {(X, E, S0 , . . . , Sd ) | (X, E) ∈ K & S0 , . . . , Sd is an edge colouring of (X, E)}, as in Example 2.2.7. By Vizing’s Theorem, every element of K admits an expansion in K∗ . This is false in the Borel context. In particular, Marks has shown that there is a d-regular acyclic Borel bipartite graph with Borel edge-chromatic number 2d − 1 [Mar16], and in [CJMST20] it is shown that there are counter-examples even for hyperfinite graphs. On the other hand, Vizing’s Theorem holds in the Borel setting for d = 2 [KST99] and for graphs of subexponential growth [BD25], in the measurable setting [GP20; Gre25], and in the Baire-measurable setting for bipartite graphs [BW23]. To summarize, we have the following: Theorem 2.4.21. 1. [CJMST20] For d ≥ 3, (K, K∗ ) enforces smoothness. 2. [KST99] For d = 2, every CBER is Borel expandable for (K, K∗ ). 62 3. [BD25] Let K0 ⊆ K be the subclass of graphs of subexponential growth. Then every CBER is Borel expandable for (K0 , K∗ ). 4. [BW23] Let K1 ⊆ K be the subclass of bipartite graphs. Then every CBER is generically expandable for (K1 , K∗ ). In particular, K1 admits Γ-equivariant expansions to K∗ generically for every countably infinite group Γ. 5. [GP20; Gre25] Every CBER is a.e. expandable for (K, K∗ ) for every (not necessarily invariant) probability Borel measure. In particular, for every countably infinite group Γ: a) there is a Borel invariant set Z ⊆ F r(K(Γ)) which admits a Borel Γequivariant expansion to K∗ and such that every invariant random Kstructure µ on Γ which concentrates on F r(K(Γ)) satisfies µ(Z) = 1; and b) Γ admits random expansions from K to K∗ . Proof. By [CJMST20, Theorem 1.4], we may fix an aperiodic hyperfinite CBER E and Borel d-regular acyclic graph G on E which does not admit a Borel edge colouring with d+1 colours. By [JKL02, Lemma 3.23], E does not admit an invariant probability Borel measure (as G is a treeing of E for which every component has infinitely-many ends), so by Nadkarni’s Theorem E is compressible. (1) follows by Proposition 2.3.25. For the “in particular” parts of (4), (5), note that F r(K1 (Γ)) is dense Gδ in K1 (Γ) and apply Proposition 2.3.4. For (5a), apply the proof of [GP20, Theorem 4.3] to the canonical K-structuring of F r(K(Γ)), as in the proof of (1) above. Remark 2.4.22. When d = 2, the same construction as in the proof of Theorem 2.4.15 gives an analogous characterization of exactly when there is a Borel equivariant expansion map from Z ⊆ K(Γ) to K∗ . We note also that (K, K∗ ) enforces smoothness even if we restrict K to the class of n-regular acyclic bipartite graphs (with a given bipartition), for n > d/2 + 1, by [CJMST20, Theorem 1.4]. Finally, we remark that the proof of the main result of [GP20] (along with Nadkarni’s Theorem) gives the stronger fact that for every CBER E on X and Borel K-structuring A of E, there is a Borel E-invariant set C so that AC admits a Borel expansion to K∗ and E(X \ C) is compressible. 63 In the Baire-measurable setting, Qian and Weilacher have shown that if we replace K∗ with (d + 2)-edge colourings, then every CBER is generically expandable [QW22]. It is open whether every CBER is generically expandable for (K, K∗ ). 2.4.8 Matchings As in Example 2.2.8, let K = {(X, E) | (X, E) is a connected, bipartite, locally finite graph satisfying Hall’s Condition}, K∗ = {(X, E, M ) | (X, E) ∈ K & M ⊆ E is a perfect matching}. All graphs below are assumed to be in K, unless specified otherwise. By Hall’s Theorem, every element of K admits an expansion in K∗ . This is false in the Borel context. Laczkovich and Conley and Kechris have given examples of d-regular hyperfinite graphs with Borel chromatic number 2 which do not admit Borel perfect matchings, even generically or a.e., for d even [Lac88; CK13]. Marks later showed that there are d-regular, acyclic graphs with Borel chromatic number 2 that do not have Borel perfect matchings for all d ≥ 2 [Mar16], and in [CJMST20, Theorem 1.4] this was extended to hyperfinite graphs. Kun has given examples of such graphs that are not hyperfinite and do not admit Borel perfect matchings a.e. [Kun24], and in [BKS22] a hyperfinite one-ended bounded-degree graph with Borel chromatic number 2 is constructed which does not admit a Borel perfect matching a.e. On the other hand, if we strengthen our structural assumptions on the graphs one can guarantee the existence of Borel perfect matchings generically or a.e. For instance, Marks and Unger have shown that if we strengthen Hall’s Condition to assume that |N (A)| ≥ (1 + ε)|A| for some fixed ε > 0 then there is always a Borel perfect matching generically [MU16], and Lyons and Nazarov have shown that Borel perfect matchings exist a.e. for graphs that instead satisfy an analogous expansion property for measure [LN11]. Conley and Miller have shown that acyclic graphs of minimum degree at least 2 which do not have infinite injective rays of degree 2 on even vertices have Borel perfect matchings generically, and a.e. when the graph is hyperfinite [CM17] (they showed this even for locally countable graphs in the measurable setting). Bowen, Kun, and Sabok have shown that Borel perfect matchings exist a.e. for hyperfinite measure-preserving regular graphs that are one-ended or have odd degree [BKS22], and in [BCW24] the odd-degree case is shown to hold even when the measure is not preserved. Borel perfect matchings also exist generically for regular graphs that are one-ended [BPZ24] 64 or have odd degree [BCW24], and for bounded-degree non-amenable vertex-transitive graphs [KL23] (note that this last result applies to all graphs, not just those in K). We note also that Borel perfect matchings have been shown to exist a.e. for some Schreier graphs of free actions of groups; see e.g. [LN11; MU16; CL17; GMP17; BKS22; GJKS24; Wei24] and [KM20, Sections 14, 15]. Remark 2.4.23. Let G be any graph (not necessarily in K). A fractional perfect matching on G is an assignment to each edge of G a weight in [0, 1] so that for every vertex v in G, the sum of the weights of the edges incident to v is equal to 1. Perfect matchings are then the same as {0, 1}-valued fractional perfect matchings. We say a fractional perfect matching is non-integral if it takes values in (0, 1). The general strategy employed by [BKS22; BPZ24; BCW24] to find Borel perfect matchings in a Borel locally finite graph G is to start with a Borel non-integral fractional perfect matching on G, and then to attempt to round this Borel fractional perfect matching to be {0, 1}-valued (off of a meagre or null set). When G is d-regular there is always a Borel non-integral fractional perfect matching on G, namely the one giving weight to 1/d to every edge. However, Borel non-integral fractional perfect matchings can also be shown to exist (possibly off of a meagre or null set) in other contexts; see [Tim23] for an example of this in the measurable setting. The results of these papers can therefore be applied to a larger class of graphs than e.g. the regular ones. It may therefore be interesting to consider separately the expansion problems for (K, K0 ) and (K0 , K∗ ), where K0 is the class of graphs equipped with a (non-integral) fractional perfect matching, though we do not explore this here. We summarize a few of the aforementioned results below, in the language and setting of expansions. Let Kd (resp. Kd,ac ) denote the subclass of K consisting of d-regular (resp. d-regular acyclic) graphs. Note that these are Gδ classes of structures. Let K0 ⊆ K denote the class of graphs that are either acyclic with no infinite injective rays of degree 2 on even vertices, are regular and one-ended, or are regular of odd degree. Let K1 ⊆ K denote the class of graphs that satisfy the strengthening of Hall’s Condition for ε-expansion for some ε > 0, or are vertex-transitive, non-amenable and have bounded degree. These are Borel classes of structures. Theorem 2.4.24. 65 1. [CJMST20] (Kd,ac , K∗ ) enforces smoothness for d ≥ 2. [Lac88; CK13] In particular, (Kd , K∗ ) and (K, K∗ ) enforce smoothness. 2. K2 does not admit Γ-equivariant expansions to K∗ generically, for any countably infinite group Γ. 3. For every countably infinite group Γ and d > 2, Kd admits Γ-equivariant expansions to K∗ generically. [MU16] So does Kd,ac . 4. [MU16; CM17; BPZ24; BCW24; KL23] Every CBER is generically expandable for (K0 ∪ K1 , K∗ ). 5. [CM17; BKS22] Every hyperfinite CBER is a.e. expandable for (K0 , K∗ ) for every invariant probability Borel measure. In particular, for every countably infinite amenable group Γ: a) there is a Borel invariant set Z ⊆ F r(K0 (Γ)) which admits a Borel Γequivariant expansion to K∗ and such that every invariant random K0 structure on Γ which concentrates on F r(K0 (Γ)) satisfies µ(Z) = 1; and b) Γ admits random expansions from K0 to K∗ . Proof. (1) By [CJMST20, Theorem 1.4], there is an aperiodic hyperfinite CBER and a Borel d-regular acyclic graph G on E which does not admit a Borel perfect matching. By [JKL02, Lemma 3.23], E does not admit an invariant probability Borel measure (as G is a treeing of E for which every component has infinitely-many ends), so by Nadkarni’s Theorem E is compressible. We then apply Proposition 2.3.25. (2) Suppose otherwise, and let X ⊆ K2 (Γ) be Borel, comeagre and invariant, and let f : X → K∗ (Γ) be a Borel equivariant expansion. It is not hard to see that the set of all A for which for all A0 ∈ AgeΓ (K2 ) there is some γ ∈ Γ with γA0 v A is a dense Gδ set in K2 (Γ), and we may therefore assume that every element of X has this property. By further shrinking X, we may assume that f is continuous. Fix now some A ∈ X and γ0 , γ1 ∈ Γ so that γ0 , γ1 are matched in f (A). By continuity and equivariance, there is some A0 ∈ AgeΓ (K2 ) so that A0 v A, and whenever γA0 v B ∈ X we have that γγ0 , γγ1 are matched in f (B). Let A1 ∈ AgeΓ (K2 ) and γ ∈ Γ be such that A0 , γA0 v A1 , A1 is connected, and the unique path in A1 whose first edge is {γ0 , γ1 } and whose last edge is {γγ0 , γγ1 } has 66 even length. Let δ be such that δA1 v A. Then {δγ0 , δγ1 }, {δγγ0 , δγγ1 } ∈ f (A), but the unique path in A containing these edges at either end has even length, which is impossible as A is a bi-infinite line and f (A) is a perfect matching. (3) By [BPZ24, Theorem 1.2], it suffices by Proposition 2.3.4(2) to show that the generic element of Kd is one-ended (note that F r(Kd (Γ)) is comeagre in Kd (Γ)). To see this, note that a d-regular graph G is one-ended if and only if for every finite set F of vertices and all vertices u, v, one of the following holds: • There is some finite set of vertices F 0 such that at least one of u, v is contained in F 0 , and the boundary of F 0 is contained in F . • There is a path in G from u to v which does not include any vertices in F . It is easy to see that the set of graphs satisfying one of these conditions for any fixed F, u, v is open and dense in Kd , and hence the set of graphs satisfying these conditions for all F, u, v is comeagre. For Kd,ac , this follows by Proposition 2.3.4(2) and [MU16, Theorem 1.3]. (4) is an immediate consequence of the (proofs in) the cited papers, and (5b) follows similarly by Proposition 2.3.4(3). (5a) We split K0 into three parts: The acyclic graphs with no infinite injective rays of degree 2 on even vertices, the regular one-ended graphs, and the regular odd-degree graphs. We will give some detail for the last case, and then sketch the first two. For the regular odd-degree graphs, we argue as follows: We consider each degree d ≥ 3 separately. Let X be the free part of K0 (Γ) restricted to the regular d-degree graphs and let A be the canonical structuring of X. By the proof of [BKS22, Theorem 1.3] one can associate to each t ∈ 2N a Borel fractional perfect matching on A so that for every invariant probability Borel measure µ on X, for almost every t the corresponding fractional perfect matching is {0, 1}-valued for µ-a.e. component of A. By [Kec95, 18.6], we can choose in a uniformly Borel way a Borel fractional perfect matching fµ for every ergodic invariant measure µ on X, so that fµ is {0, 1}-valued for µ-a.e. component of A. Let Xµ denote the set of components for which fµ is {0, 1}-valued. By considering S an ergodic decomposition of X (cf. [Kec25, Theorem 5.12]), the set Z = µ Xµ is S Borel, and f = µ fµ Xµ gives a Borel perfect matching of AZ. Moreover, µ(Z) = 1 for every invariant probability Borel measure on X. By Proposition 2.3.4(2), we are done. 67 For the acyclic graphs with no infinite injective rays of degree 2 on even vertices, the argument is similar: We consider an ergodic decomposition, and note that the proof of [CM17, Theorem B] is effective enough that the union of the solutions (and their domains) for all ergodic invariant probability Borel measures is still Borel. For the regular one-ended graphs, we again consider an ergodic decomposition and argue that the proof of [BKS22, Theorem 1.1] is sufficiently uniform. For a fixed measure µ, the proof proceeds by constructing a transfinite sequence of fractional perfect matchings, and arguing that this must stabilize at some countable ordinal. We claim that the construction of this sequence is effective (in µ). Then, by the Boundedness Theorem for analytic well-founded relations [Kec95, 31.1] there is a uniform bound on how long these sequences take to stabilize for all (ergodic) invariant probability Borel measures, so we are done by the same argument as in the previous two cases. The verification that the construction is effective is tedious but straightforward. The most subtle step is in the use of the Choquet–Bishop–de Leeuw Theorem, which is sufficiently effective for separable metrizable spaces as this essentially boils down to an application of compact uniformization; see e.g. [Phe01, Section 3; Sim09, IV.9; Kec95, 28.8]. 2.5 Problems Problem 2.5.1. If (K, K∗ ) satisfies the hypotheses of Theorem 2.3.13 and admits generic equivariant expansions, is every CBER generically expandable for (K, K∗ )? Problem 2.5.2. Does the conclusion of Theorem 2.3.13 hold for classes of structures without TAC? In [CK18], it is shown that for many natural classes of aperiodic CBER E, there is a Borel class of structures K so that E ∈ E if and only if E admits a Borel K-structuring. Nonetheless, it is interesting whether there is any “natural” class of problems (e.g. problems that are studied in finite combinatorics) that carve out interesting classes E of CBER. Example 2.2.6 was an attempt to characterize hyperfiniteness, though we have seen that it actually enforces smoothness. Problem 2.5.3. Let E be a class of aperiodic CBER such as those that are hyperfinite, (non)-compressible or treeable. Is there a “natural” expansion problem (K, K∗ ) for which an aperiodic CBER E is Borel expandable if and only if E ∈ E? 68 In [GX24] a problem is described for which E admits solutions exactly when E is hyperfinite. However, this does not fit the framework of expansion problems, as it involves finding “approximate” solutions. We note that the spanning tree example (Example 2.2.5) corresponds to a class of CBER that lies somewhere between hyperfinite and treeable. Problem 2.5.4. What is the class of CBER that are Borel expandable for the spanning tree problem? In general, it would be interesting to answer the problems remaining in Table 2.1. We highlight a few of these below. Problem 2.5.5. For the Ramsey expansion problem (Example 2.2.2), when an invariant random structure admits an invariant random expansion? Can we characterize a.e. expansions (in the sense of Section 2.3.3)? Under what assumptions to generic expansions exist on CBER? Problem 2.5.6. Can we say more about when a Borel structuring of a CBER by partial orders is expandable to a Borel structuring by linear orders? In particular, is there a characterization of exactly which invariant random expansions come from push-forwards along a.e. equivariant expansion maps? Problem 2.5.7. Does Vizing’s Theorem hold generically? There are many open problems regarding the existence of perfect matchings; see Section 2.4.8 for details. As noted in Remark 2.4.23, one can often find perfect matchings by first finding non-integral fractional perfect matchings, and then rounding them. Problem 2.5.8. What can be said about the expansion problem of finding a nonintegral fractional perfect matching on a Borel graph? When can we round non-integral fractional perfect matchings to perfect matchings? See e.g. [BKS22; Tim23; BPZ24] for some partial results and examples. 2.A Existence of frequencies The purpose of this appendix is to prove Lemma 2.4.19 on the existence of frequencies. The proof is essentially the same as that of [LS99, Lemma 4.2], though we work here 69 in a more general setting; we also thank Minghao Pan for sharing with us his notes about this proof. We begin by recalling some definitions. Let Γ be a countably infinite group and let (Zn )n∈N be a random walk on Γ with symmetric step distribution µ whose support generates Γ. (Note that we do not assume µ to be finitely supported.) For γ ∈ Γ, let Pγ denote the law of the random walk (Zn )n∈N starting at γ. Define Ω(Γ, µ) to be the set of all W ⊆ Γ for which there exists r ∈ [0, 1] so that lim n→∞ X 1 n−1 1(Zi ∈ W ) = r, Pγ -a.s. for all γ ∈ Γ, n i=0 and for W ∈ Ω(Γ, µ) we let Freqµ (W ) be the unique such r. We note that Ω(Γ, µ) ⊆ 2Γ , Freqµ : Ω(Γ, µ) → [0, 1] are Borel and Γ-invariant. (Here, 1(Zn ∈ W ) is equal to 1 when Zn ∈ W and 0 otherwise.) Note that if limn→∞ n1 n−1 i=0 1(Zi ∈ W ) converges Pγ -almost surely for some γ, then it does for all γ. To see this, note that the support of µ generates Γ, so if the sequence diverges with positive probability for a random walk starting at some γ, then this happens with positive probability for a random walk starting at any γ. Similarly, we see that the value of the limit (should it exist) does not depend on the choice of γ. P Let now K denote the class of equivalence relations. Let ν be an invariant random equivalence relation on Γ, i.e. an invariant random K-structure on Γ, and let E ∼ ν. Let P e denote the identity in Γ. We will show that ν-almost surely, limn→∞ n1 n−1 i=0 1(Zi ∈ C) converges to a constant value Pe -a.s. for every E-class C. By the previous remark, this proves Lemma 2.4.19. A two-sided random walk starting at γ is a sequence of random variables (Zn )n∈Z so that (Zn )n∈N and (Z−n )n∈N are random walks starting at γ. Let P̂γ denote the law of the two-sided random walk starting at γ. Note that Γ acts on ΓZ by coordinate-wise multiplication, so that we may consider K(Γ) × ΓZ with the diagonal action of Γ: γ · (E, (Zn )n∈Z ) = (γ · E, (γ · Zn )n∈Z ). We also define the shift map S : K(Γ)×ΓZ → K(Γ)×ΓZ to be the map S(E, (Zn )n∈Z ) = (E, (Zn+1 )n∈Z ). Note that the actions of Γ, S on K(Γ) × ΓZ commute. 70 Let I denote the σ-algebra of Γ-invariant Borel sets in K(Γ) × ΓZ , and set λ = ν × P̂e . Also, for E ∈ K(Γ), let Γ/E denote the set of equivalence classes of E. Claim 2.A.1. If A ∈ I, then λ(A) = λ(S · A). Proof. Let Wγn = {Z = (Zn )n∈Z : Zn = γ}. Note that by the symmetry of µ, X P̂γ [Wγjj ∩ · · · ∩ Wγkk ] = k−1 Y µ(γi−1 γi+1 ) i=j γ∈Γ for all j < 0 < k ∈ Z and γj , . . . , γk ∈ Γ. It follows that Let now κ = ν × we have γ∈Γ P̂γ is shift-invariant. P γ∈Γ P̂γ . Note that κ is Γ-invariant. If A ∈ I, then by Γ-invariance P κ(A ∩ We0 ) = X κ(A ∩ We0 ∩ Wγ−1 ) = γ X κ(A ∩ Wγ0 ∩ We−1 ) = κ(A ∩ We−1 ). γ It follows that λ(SA) = κ(SA ∩ We0 ) = κ(SA ∩ We−1 ) = κ(S(A ∩ We0 )) = κ(A ∩ We0 ) = λ(A). / For C ⊆ Γ, Z = (Zn )n∈Z ∈ ΓZ , m < n ∈ Z, let n αm (C, Z) = X 1 n−1 1(Zi ∈ C), n − m i=m and for (E, Z) ∈ K(Γ) × ΓZ , n, k ∈ N let Fkn (E, Z) = n·max{α0n (C0 , Z)+· · ·+α0n (Ck−1 , Z) : C0 , . . . , Ck−1 are distinct E-classes}. It is easy to see that Fki+j (E, Z) ≤ Fki (E, Z) + Fkj (S i · (E, Z)) and that each Fkn is Γ-invariant. By Claim 2.A.1 and Kingman’s Subadditive Ergodic Theorem (see e.g. F n (E,Z) [Ste89]) there are Γ, S-invariant maps Fk , k ∈ N so that Fk (E, Z) = limn→∞ k n λ-a.s. Let now An (E, Z) = {|α0m (C, Z) − α0k (C, Z)| : k, m ≥ n & C ∈ Γ/E}. Claim 2.A.2. limn→∞ max(An (E, Z)) = 0 for almost every (E, Z). 71 Proof. Fix (E, Z) for which Fk (E, Z) = limn→∞ Fkn (E,Z) n for all k ∈ N. F n (C,Z) Fn For any E-class C and n ∈ N, there is some k so that α0n (C, Z) = k n − k−1n , namely the k for which C is the k-th most frequently visited E-class in the first n steps of Z. It follows that ( α0m (C, Z) ∈ Sn = m (E, Z) Fkm (E, Z) Fk−1 − : k ≥ 1, m ≥ n m m (C,Z) ) for m ≥ n. Let Sn (δ) denote the δ-neighbourhood of Sn in [0, 1]. Since |α0m+1 (C, Z) − α0m (C, Z)| ≤ 1 , the set {α0m (C, Z) : m ≥ n} is contained in a single connected component of Sn ( n1 ), m for every E-class C. It therefore suffices to show that for all ε > 0, there is some n sufficiently large that Sn ( n1 ) has length at most ε. Fix now ε > 0, and fix k sufficiently large that Fj (E, Z) − Fj−1 (E, Z) ≤ ε for j ≥ k. F m (E,Z) ε Fix n so that |Fj (E, Z) − j m | ≤ k+1 for all j ≤ k and m ≥ n. It follows that ε every point in Sn is within distance ε from 0, or k+1 from Fj (E, Z) − Fj−1 (E, Z) for some j ≤ k, so that Sn ( n1 ) has length at most 3(ε + n1 ). Since ε was arbitrary, this proves the claim. / It follows that for almost every (E, Z), (α0n (C, Z))n∈N is Cauchy for every E-class C, 0 and hence this sequence converges. Symmetrically, (α−n (C, Z))n∈N converges a.s. for every E-class C. Note that with probability 1 n→∞ max |αn2n (C, Z) − α0n (C, Z)| = 2 · max |α02n (C, Z) − α0n (C, Z)| −−−→ 0, C∈Γ/E C∈Γ/E so we may fix a sequence nk such that h i P max |αn2nk k (C, Z) − α0nk (C, Z)| ≥ 2−k ≤ 2−k . C∈Γ/E By Claim 2.A.1, h i h i 0 P max |αn2nk k (C, Z) − α0nk (C, Z)| ≥ 2−k = P max |α0nk (C, Z) − α−n (C, Z)| ≥ 2−k , k C C so by the Borel–Cantelli Lemma we have that for almost every (E, Z), 0 max |α0nk (C, Z) − α−n (C, Z)| < 2−k k C∈Γ/E for all but finitely many k. It follows that 0 lim α0n (C, Z) = lim α−n (C, Z) n→∞ n→∞ 0 almost surely for all C ∈ Γ/E. But for any fixed C ⊆ Γ, α0n (C, Z), α−n (C, Z) are independent, so the limits are independent and hence must be constant a.s. 72 Chapter 3 INVARIANT UNIFORMIZATION Alexander S. Kechris and Michael S. Wolman 3.1 3.1.1 Introduction Invariant uniformization and smoothness Given sets X, Y and P ⊆ X × Y with projX (P ) = X, a uniformization of P is a function f : X → Y such that ∀x ∈ X((x, f (x)) ∈ P ). If now E is an equivalence relation on X, we say that P is E-invariant if x1 Ex2 =⇒ Px1 = Px2 , where Px = {y : (x, y) ∈ P } is the x-section of P . Equivalently this means that P is invariant under the equivalence relation E × ∆Y on X × Y , where ∆Y is the equality relation on Y . In this case an E-invariant uniformization is a uniformization f such that x1 Ex2 =⇒ f (x1 ) = f (x2 ). Also if E, F are equivalence relations on sets X, Y , resp., a homomorphism of E to F is a function f : X → Y such that x1 Ex2 =⇒ f (x1 )F f (x2 ). Thus an invariant uniformization is a uniformization that is a homomorphism of E to ∆Y . Consider now the situation where X, Y are Polish spaces and P is a Borel subset of X × Y . In this case standard results in descriptive set theory provide conditions which imply the existence of Borel uniformizations. These fall mainly into two categories, see [Kec95, Section 18]: “small section” and “large section” uniformization results. We will concentrate here on the following standard instances of these results: Theorem 3.1.1 (Measure uniformization). Let X, Y be Polish spaces, µ a probability Borel measure on Y and P ⊆ X × Y a Borel set such that ∀x ∈ X(µ(Px ) > 0). Then P admits a Borel uniformization. Theorem 3.1.2 (Category uniformization). Let X, Y be Polish spaces and P ⊆ X×Y a Borel set such that ∀x ∈ X(Px is non-meager). Then P admits a Borel uniformization. Theorem 3.1.3 (Kσ uniformization). Let X, Y be Polish spaces and P ⊆ X × Y a Borel set such that ∀x ∈ X(Px is non-empty and Kσ ). Then P admits a Borel uniformization. A special case of Theorem 3.1.3 is the following: 73 Theorem 3.1.4 (Countable uniformization). Let X, Y be Polish spaces and P ⊆ X ×Y a Borel set such that ∀x ∈ X(Px is non empty and countable). Then P admits a Borel uniformization. Suppose now that E is a Borel equivalence relation on X and P in any one of these results is E-invariant. When does there exist a Borel E-invariant uniformization, i.e., a Borel uniformization that is also a homomorphism of E to ∆Y ? We say that E satisfies measure (resp., category, Kσ , countable) invariant uniformization if for every Y, µ, P as in the corresponding uniformization theorem above, if P is moreover E-invariant, then it admits a Borel E-invariant uniformization. The following gives a complete answer to this question. Recall that a Borel equivalence relation E on X is smooth if there is a Polish space Z and a Borel function S : X → Z such that x1 Ex2 ⇐⇒ S(x1 ) = S(x2 ). Theorem 3.1.5. Let E be a Borel equivalence relation on a Polish space X. Then the following are equivalent: (i) E is smooth; (ii) E satisfies measure invariant uniformization; (iii) E satisfies category invariant uniformization; (iv) E satisfies Kσ invariant uniformization; (v) E satisfies countable invariant uniformization. One can compute the exact definable complexity of counterexamples to invariant uniformization. Let E0 denote the non-smooth Borel equivalence relation on 2N given by xE0 y ⇐⇒ ∃m∀n ≥ m(xn = yn ). In the proof of Theorem 3.1.5, it is shown that for E = E0 on X = 2N we have the following: (1) Failure of measure invariant uniformization: There are Y, µ, E-invariant P ∈ Fσ with µ(Px ) = 1, for all x ∈ X, which has no Borel E-invariant uniformization. (2) Failure of category invariant uniformization: There is Y and an E-invariant Q ∈ Gδ with Qx comeager, for all x ∈ X, which has no Borel E-invariant uniformization. (3) Failure of countable invariant uniformization: There is Y and an E-invariant P ∈ Fσ such that Px is non-empty and countable, for all x ∈ X, which has no Borel E-invariant uniformization. 74 The definable complexity of Q, P in (2), (3) is optimal. In the case of measure invariant uniformization, however, there are counterexamples which are Gδ , and this together with (1) gives the optimal definable complexity of counterexamples to measure invariant uniformization. These results are the contents of Theorems 3.1.6 and 3.1.7. Theorem 3.1.6. Let X ⊆ 2N be the sequences with infinitely many ones. There is a Polish space Y , a probability Borel measure µ on Y and an E0 -invariant Gδ set P ⊆ X × Y with Px comeager and µ(Px ) = 1, for all x ∈ X, which has no Borel E0 -invariant uniformization. Theorem 3.1.7. Let X, Y be Polish spaces, E a Borel equivalence relation on X and P ⊆ X × Y an E-invariant Borel relation. Suppose one of the following holds: (i) Px ∈ ∆02 and µx (Px ) > 0, for all x ∈ X, and some Borel assignment x 7→ µx of probability Borel measures µx on Y ; (ii) Px ∈ Fσ and Px non-meager, for all x ∈ X; (iii) Px ∈ Gδ and Px non-empty and Kσ (in particular countable), for all x ∈ X. Then there is a Borel E-invariant uniformization. The proof of Theorem 3.1.6 uses the Ramsey property. 3.1.2 Local dichotomies The equivalence of (i) and (v) in Theorem 3.1.5 essentially reduces to the fact that if E is a countable Borel equivalence relation (i.e., one for which all of its equivalence classes are countable) which is not smooth, then the relation (x, y) ∈ P ⇐⇒ xEy, is clearly E-invariant with countable nonempty sections but has no E-invariant uniformization. Considering the problem of invariant uniformization “locally”, Miller [Mild] recently proved the following dichotomy that shows that this is essentially the only obstruction to (v). Below E0 × IN is the equivalence relation on 2N × N given by (x, m)E0 × IN (y, n) ⇐⇒ xE0 y. Also if E, F are equivalence relations on spaces X, Y , resp., an embedding of E into F is an injection π : X → Y such that x2 Ex2 ⇐⇒ π(x1 )F π(x2 ). 75 Theorem 3.1.8 ([Mild, Theorem 2]). Let X, Y be Polish spaces, E a Borel equivalence relation on X and P ⊆ X × Y an E-invariant Borel relation with countable non-empty sections. Then exactly of the following holds: (1) There is a Borel E-invariant uniformization, (2) There is a continuous embedding πX : 2N × N → X of E0 × IN into E and a continuous injection πY : 2N × N → Y such that for all x, x0 ∈ 2N × N, ¬(x E0 × IN x0 ) =⇒ PπX (x) ∩ PπX (x0 ) = ∅ and PπX (x) = πY ([x]E0 ×IN ). We provide a different proof of this dichotomy, using Miller’s (G0 , H0 ) dichotomy [Mil12] and Lecomte’s ℵ0 -dimensional hypergraph dichotomy [Lec09]. Our proof relies on the following strengthening of (i) =⇒ (v) of Theorem 3.1.5, which is interesting in its own right: Theorem 3.1.9. Let F be a smooth Borel equivalence relation on a Polish space X, Y be a Polish space, and P ⊆ X × Y be a Borel set with countable sections. Suppose that \ Px 6= ∅ x∈C for every F -class C. Then P admits a Borel F -invariant uniformization. We also prove an ℵ0 -dimensional (G0 , H0 )-type dichotomy, which generalizes Lecomte’s dichotomy in the same way that the (G0 , H0 ) dichotomy generalizes the G0 dichotomy, and use this to give still another proof of Theorem 3.1.8. In the case of countable uniformization, the Lusin-Novikov theorem asserts that P can be covered by the graphs of countably-many Borel functions. When E is smooth, the proof of Theorem 3.1.5 gives an invariant analogue of this fact (cf. Theorem 3.2.3). De Rancourt and Miller [dRM] have shown that E0 is essentially the only obstruction to invariant Lusin-Novikov: Theorem 3.1.10 ([dRM, Theorem 4.11]). Let X, Y be Polish spaces, E a Borel equivalence relation on X and P ⊆ X × Y an E-invariant Borel relation with countable non-empty sections. Then exactly one of the following holds: (1) There is a sequence gn : X → Y of Borel E-invariant uniformizations with S P = n graph(gn ). 76 (2) There is a continuous embedding πX : 2N → X of E0 into E and a continuous injection πY : 2N → Y such that for all x ∈ 2N , P (πX (x), πY (x)). We provide a different proof of this theorem in Section 3.4.4, directly from Miller’s (G0 , H0 ) dichotomy. 3.1.3 Anti-dichotomy results Our next result can be viewed as a sort of anti-dichotomy theorem for large-section invariant uniformizations (see also the discussion in [TV21, Section 1]). Informally, dichotomies such as Theorem 3.1.8 provide upper bounds on the complexity of the collection of Borel sets satisfying certain combinatorial properties. Thus, one method of showing that there is no analogous dichotomy is to provide lower bounds on the complexity of such sets. In order to state this precisely, we first fix a “nice” parametrization of the Borel relations on NN , i.e., a Π11 set D ⊆ 2N and a map D 3 d 7→ Dd such that each Dd ⊆ NN × NN , d ∈ D is Borel, each Borel set in NN appears as some Dd , and so that these satisfy some natural definability properties (cf. [AK00, Section 5]). Define now P = {(d, e) : Dd is an equivalence relation on NN and De is Dd -invariant}, and let P unif denote the set of pairs (d, e) ∈ P for which De admits a Dd -invariant uniformization. More generally, for any set A of properties of sets P ⊆ NN × NN , let PA (resp. PAunif ) denote the set of pairs (d, e) in P (resp. P unif ) such that De unif satisfies all of the properties in A. Let Pctble (resp. Pctble ) denote PA (resp. PAunif ) for A consisting of the property that P has countable sections. One can easily check that P is Π11 and that P unif is Σ12 . The same is true for Pctble unif and Pctble . In the latter case, however, the effective version of Theorem 3.1.8 (see Theorem 3.4.14) gives a better bound on the complexity: unif Proposition 3.1.11. The set Pctble is Π11 . By contrast, in the case of large sections, we prove the following, where a set B in a Polish space X is called Σ12 -complete if it is Σ12 , and for all zero-dimensional Polish spaces Y and Σ12 sets C ⊆ Y there is a continuous function f : Y → X such that C = f −1 (B). 77 Theorem 3.1.12. The set PAunif is Σ12 -complete, where A is one of the following sets of properties of P ⊆ NN × NN : 1. P has non-meager sections; 2. P has non-meager Gδ sections; 3. P has non-meager sections and is Gδ ; 4. P has µ-positive sections for some probability Borel measure µ on NN ; 5. P has µ-positive Fσ sections for some probability Borel measure µ on NN ; 6. P has µ-positive sections for some probability Borel measure µ on NN and is Fσ . The same holds for comeager instead of non-meager, and µ-conull instead of µ-positive. In fact, there is a hyperfinite Borel equivalence relation E with code d ∈ D such that for all such A above, the set of e ∈ D such that (d, e) ∈ PAunif is Σ12 -complete. Problem 3.1.13. Is there an analogous dichotomy or anti-dichotomy result for the case where P has Kσ sections? While we do not know the answer to this problem, we note that Theorem 3.1.9 is false when the sections are only assumed to be Kσ : Proposition 3.1.14. There is a smooth countable Borel equivalence relation F on NN and an open set P ⊆ NN × 2N such that \ Px 6= ∅ x∈C for every F -class C, but which does not admit a Borel F -invariant uniformization. 3.1.4 Invariant countable uniformization We next consider a somewhat less strict notion of invariant uniformization, where instead of selecting a single point in each section we select a countable nonempty subset. More precisely, given Polish spaces X, Y , a Borel equivalence relation E on X and an E-invariant Borel set P ⊆ X × Y , with projX (P ) = X, a Borel Einvariant countable uniformization is a Borel function f : X → Y N such that ∀x ∈ X∀n ∈ N((x, f (x)n ) ∈ P ) and x1 Ex2 =⇒ {f (x1 )n : n ∈ N} = {f (x2 )n : n ∈ N}. 78 Equivalently, if for each Polish space Y , we denote by EYctble the equivalence relation on Y N given by (xn )EYctble (yn ) ⇐⇒ {xn : n ∈ N} = {yn : n ∈ N}, then an E-invariant countable uniformization is a Borel homomorphism f of E to EYctble such that for each x, n, we have that (x, f (x)n ) ∈ P . We say that E satisfies measure (resp., category, Kσ ) countable invariant uniformization if for every Y, µ, P as in the corresponding uniformization theorem above, if P is moreover E-invariant, then it admits a Borel E-invariant countable uniformization. Recall that a Borel equivalence relation E on X is reducible to countable if there is a Polish space Z, a countable Borel equivalence relation F on Z and a Borel function S : X → Z such that x1 Ex2 ⇐⇒ S(x1 )F S(x2 ). As in the proof below of Theorem 3.1.5, part (A), one can see that if a Borel equivalence relation E on X is reducible to countable, then E satisfies measure (resp. category, Kσ ) countable invariant uniformization. We conjecture the following: Conjecture 3.1.15. Let E be a Borel equivalence relation on a Polish space X. Then the following are equivalent: (a) E is reducible to countable; (b) E satisfies measure countable invariant uniformization; (c) E satisfies category countable invariant uniformization; (d) E satisfies Kσ countable invariant uniformization. We discuss some partial results in Section 3.5. 3.1.5 Further invariant uniformization results and smoothness We have so far considered the existence of Borel invariant uniformizations, generalizing the standard “small section” and “large section” uniformization theorems. One can also consider invariant analogues of uniformization theorems for more general pointclasses, such as the following: Theorem 3.1.16 (Jankov, von Neumann uniformization [Kec95, 18.1]). Let X, Y be Polish spaces and P ⊆ X × Y be a Σ11 set such that Px is non-empty, for all x ∈ X. Then P has a uniformization function which is σ(Σ11 )-measurable. 79 Theorem 3.1.17 (Novikov-Kondô uniformization [Kec95, 36.14]). Let X, Y be Polish spaces and P ⊆ X × Y be a Π11 set such that Px is non-empty, for all x ∈ X. Then P has a uniformizatoin function whose graph is Π11 . Let E be a Borel equivalence relation on X. We say E satisfies Jankov-von Neumann (resp. Novikov-Kondô) invariant uniformization if for every Y, P as in the corresponding uniformization theorem above, if P is moreover E-invariant, then it admits an E-invariant uniformization which is definable in the same sense as in the corresponding uniformization theorem. The following characterization of those Borel equivalence relations that satisfy these properties essentially follows from the proof of Theorem 3.1.5. Theorem 3.1.18. Let E be a Borel equivalence relation on a Polish space X. Then the following are equivalent: (i) E is smooth; (ii) E satisfies Jankov-von Neumann invariant uniformization; (iii) E satisfies Novikov-Kondô invariant uniformization. 3.1.6 Remarks on invariant uniformization over products One can consider more generally the question of invariant uniformization over products. Let X, Y be Polish spaces, E a Borel equivalence on X, F a Borel equivalence on Y , and P ⊆ X × Y an E × F -invariant set. In this case, one can ask whether there is an E × F -invariant Borel set U ⊆ P so that each section Ux intersects one, or even finitely-many, F -classes. This paper then considers the special case where F = ∆Y is equality. In the case where P has countable sections and F is smooth, one can reduce this to the case where F is equality to get analogues of Theorems 3.1.8 and 3.1.10. Miller [Mild, Theorem 2.1] has proved a generalization of Theorem 3.1.8 where P has countable sections and the equivalence classes of F are countable, and de Rancourt and Miller [dRM, Theorem 4.11] have proved a generalization of Theorem 3.1.10 where the sections of P are contained in countably many F -classes (but are not necessarily countable). The problem of invariant uniformization is also discussed in [Mye76; BM75] where they consider the question of invariant uniformization over products when E, F come 80 from Polish group actions, and specifically when E, F are the isomorphism relation on a class of structures. Myers [Mye76, Theorem 10] gives an example in which there is no Baire-measurable invariant uniformization, so that in particular the invariant Jankov-von Neumann and invariant Novikov-Kondô uniformization don’t hold. Acknowledgements. Research partially supported by NSF Grant DMS-1950475. We would like to thank Todor Tsankov who asked whether measure invariant uniformization holds for countable Borel equivalence relations. We would also like to thank Ben Miller, Andrew Marks and Dino Rossegger for useful comments and discussion. 3.2 Proof of Theorem 3.1.5 (A) We first show that (i) implies (ii), the proof that (i) implies (iii) being similar. Fix a Polish space Z and a Borel function S : X → Z such that x1 Ex2 ⇐⇒ S(x1 ) = S(x2 ). Fix also Y, µ, P as in the definition of measure invariant uniformization. Define P ∗ ⊆ Z × Y as follows:   (z, y) ∈ P ∗ ⇐⇒ ∀x ∈ X S(x) = z =⇒ (x, y) ∈ P . Then P ∗ is Π11 and we have that S(x) = z =⇒ Pz∗ = Px , z∈ / S(X) =⇒ Pz∗ = Y. Thus ∀z ∈ Z(µ(Pz∗ ) > 0). Then, by [Kec95, 36.24], there is a Borel function f ∗ : Z → Y such that ∀z ∈ Z((z, f ∗ (z)) ∈ P ∗ ). Put f (x) = f ∗ (S(x)). Then f is an E-invariant uniformization of P . We next prove that (i) implies (iv) (and therefore (v)). Fix Z, S as in the previous case and Y, P as in the definition of Kσ invariant uniformization. Define P ∗ as before. Then A = {(z, y) : ∃x ∈ X(S(x) = z & P (x, y))} is a Σ11 subset of P ∗ , so by the Lusin separation theorem there is a Borel subset P ∗∗ of P ∗ such that A ⊆ P ∗∗ . By [Kec95, 35.47], the set C of all z ∈ Z such that Pz∗∗ is Kσ is Π11 and contains the Σ11 set S(X), so by separation there is a Borel set B with A ⊆ B ⊆ C. Then if Q ⊆ Z × Y is defined by (z, y) ∈ Q ⇐⇒ z ∈ B & (z, y) ∈ P ∗∗ , 81 we have that S(x) = z =⇒ Qz = Px , and every Qz is Kσ . It follows, by [Kec95, 35.46], that D = projZ (Q) is Borel and there is a Borel function g : D → Y such that ∀z ∈ D(z, g(z)) ∈ Q. Since f (X) ⊆ D, the function f (x) = g(S(x)) is an E-invariant uniformization of P . (B) We will next show that ¬(i) implies ¬(ii), ¬(iii), and ¬(v) (and thus also ¬(iv)). We will use the following lemma. Below for Borel equivalence relations E, E 0 on Polish spaces X, X 0 , resp., we write E ≤B E 0 iff there is a Borel map f : X → X 0 such that x1 Ex2 ⇐⇒ f (x1 )E 0 f (x2 ), i.e., E can be Borel reduced to E 0 (via the reduction f ). Lemma 3.2.1. Let E, E 0 be Borel equivalence relations on Polish spaces X, X 0 , resp., such that E ≤B E 0 . If E fails (ii) (resp., (iii), (iv), (v)), so does E 0 . Proof. Let f : X → X 0 be a Borel reduction of E into E 0 . Assume first that E fails (ii) with witness Y, µ, P . Define P 0 ⊆ X 0 × Y by 0  0 0 0  (x , y) ∈ P ⇐⇒ ∀x ∈ X f (x)E x =⇒ (x, y) ∈ P . Then note that f (x)E 0 x0 =⇒ Px0 0 = Px , x0 ∈ / [f (X)]E 0 =⇒ Px0 0 = Y. Now clearly P 0 is Π11 and invariant under the Borel equivalence relation E 0 × ∆Y . Then by a result of Solovay (see [Kec95, 34.6]), there is a Π11 -rank ϕ : P 0 → ω1 which is E 0 × ∆Y -invariant. Consider then the Σ11 subset P 00 of P 0 defined by 0 (x , y) ∈ P 00  0 0  ⇐⇒ ∃x ∈ X f (x)E x & (x, y) ∈ P . By boundedness there is a Borel E 0 × ∆Y -invariant set P 000 with P 00 ⊆ P 000 ⊆ P 0 . Let now Z ⊆ X 0 be defined by 000 x0 ∈ Z ⇐⇒ µ(Px0 ) > 0. Then Z is Borel and E 0 -invariant and contains [f (X)]E 0 . Finally define Q ⊆ X 0 × Y by   (x0 , y) ∈ Q ⇐⇒ x0 ∈ Z & (x0 , y) ∈ P 000 or x0 ∈ / Z. 82 Then f (x) = x0 =⇒ Qx0 = Px , so Y, µ, Q witnesses the failure of (ii) for E 0 . The case of (iii) is similar and we next consider the case of (iv). Repeat then the previous argument for case (ii) until the definition of P 000 . Then define Z 0 ⊆ X 0 by x0 ∈ Z 0 ⇐⇒ Px0 is Kσ and nonempty. 000 Then Z 0 is Π11 , by [Kec95, 35.47] and the relativization of the fact that every nonempty ∆11 Kσ set contains a ∆11 member, see [Mos09, 4F.15]. It is also E 0 -invariant and contains [f (X)]E 0 . Let then Z be E 0 -invariant Borel with [f (X)]E 0 ⊆ Z ⊆ Z 0 and define Q as before but replacing “x0 ∈ / Z” by “(x0 ∈ / Z and y = y0 )”, for some fixed y0 ∈ Y . Then Y, Q witnesses the failure of (iv) for E 0 . Finally, the case of (v) is similar to (iv) by now defining x0 ∈ Z 0 ⇐⇒ Px0 is countable and nonempty, 000 and using that Z 0 is Π11 by [Kec95, 35.38] (and [Mos09, 4F.15] again). Assume now that E is not smooth. Then by [HKL90] we have E0 ≤B E. Thus by Lemma 3.2.1 it is enough to show that E0 fails (ii), (iii), and (v) (thus also (iv)). We first prove that E0 fails (ii). We view here 2N as the Cantor group (Z/2Z)N with pointwise addition + and we let µ be the Haar measure, i.e., the usual product measure. Let then A ⊆ (Z/2Z)N be an Fσ set which has µ-measure 1 but is meager. Let X = Y = (Z/2Z)N and define P ⊆ X × Y as follows: (x, y) ∈ P ⇐⇒ ∃x0 E0 x(x0 + y ∈ A). Clearly P is Fσ and, since Px = x0 E0 x (A − x0 ), clearly µ(Px ) = 1. Moreover P is E0 -invariant. Assume then, towards a contradiction that f is a Borel E0 -invariant uniformization. Since xE0 x0 =⇒ f (x) = f (x0 ), by generic ergodicity of E0 there is a comeager Borel E0 -invariant set C ⊆ X and y0 such that ∀x ∈ C(f (x) = y0 ); thus ∀x ∈ C(x, y0 ) ∈ P , so ∀x ∈ C∃x0 E0 x(x0 ∈ A − y0 ). If G ⊆ (Z/2Z)N is the subgroup consisting of the eventually 0 sequences, then xE0 y ⇐⇒ ∃g ∈ G(g + x = y); thus S C = g∈G (g + (A − y0 )), so C is meager, a contradiction. S To show that E0 fails (v), define (x, y) ∈ P ⇐⇒ xE0 y. Then any Borel E0 -invariant uniformization of P gives a Borel selector for E0 , a contradiction. 83 Finally to see that E0 fails (iii), use above B = (Z/2Z)N \ A, instead of A, to produce a Gδ set Q as follows: (x, y) ∈ Q ⇐⇒ ∀x0 E0 x(x0 + y ∈ B). Then Q is E0 -invariant and has comeager sections. If g is a Borel E0 -invariant uniformization, then by the ergodicity of E0 , there is a µ-measure 1 set D and y0 such that ∀x ∈ D∀x0 E0 x(x0 ∈ B − y0 ), so D ⊆ B − y0 , and thus µ(D) = 0, a contradiction. This completes the proof of Theorem 3.1.5. Remark 3.2.2. Andrew Marks and Dino Rossegger have pointed out that the construction in Proposition 3.5.8 actually gives a strengthening of Theorem 3.1.5: If E is not smooth then E fails co-countable invariant uniformization, i.e., there is an E-invariant Borel set P whose sections are co-countable which does not admit a Borel E-invariant uniformization. To see this, define a hyperfinite Borel equivalence relation E on (2N )N by xEy iff there is a permutation σ of N fixing all but finitely many numbers, so that xn = yσ(n) for n ∈ N. H. Friedman has shown the following strengthening of Theorem 3.5.6 for this equivalence relation [Fri81, Proposition C]: If F : (2N )N → 2N is Borel and E-invariant, then there is some x ∈ (2N )N such that F (x) = x0 . Let now P be as in the proof of Proposition 3.5.8. Then P is Borel, E-invariant, and has co-countable sections. If F were a Borel E-invariant uniformization of P , then there would be some x with F (x) = x0 . But by definition ¬P (x, x0 ), a contradiction. Let now E 0 be a non-smooth Borel equivalence relation on X. By [HKL90; DJK94], if E 0 is not smooth then there is a Borel reduction f from E to E 0 . Define P 0 as in the proof of Lemma 3.2.1. Then P 0 has co-countable sections and does not admit a Borel E 0 -invariant uniformization, so it remains to check that P 0 is Borel. To see this, write Q(x0 , y, x) ⇐⇒ f (x)E 0 x0 & ¬P (x, y). Then Q is Borel and its sections Q(x0 ,y) are either an E-class or empty, hence countable. Thus by Lusin–Novikov, P 0 = X × 2N \ projX×2N (Q) is Borel. (C) We note the following strengthening of Theorem 3.1.5 in the case that E is smooth, where K(Y ) denotes the Polish space of compact subsets of Y [Kec95, 4.F]: Theorem 3.2.3. Let X, Y be Polish spaces, E be a smooth Borel equivalence relation on X, and P ⊆ X × Y be a Borel E-invariant set with non-empty sections. 84 1. If P has countable sections, then P = Borel maps gn : X → Y . 2. If P has Kσ sections, then Px = maps Kn : X → K(Y ). S S n graph(gn ) for a sequence of E-invariant n Kn (x) for a sequence of E-invariant Borel 3. If P has comeager sections, then P ⊇ n Un for a sequence of E-invariant Borel sets Un ⊆ X × Y with dense open sections. Moreover, if P has dense Gδ sections, T we can find such Un with P = n Un . T Proof. The first two assertions follow from [Kec95, 18.10, 35.46] applied to Q from the proof of (i) =⇒ (iv) of Theorem 3.1.5. For the third, let Z, S, P ∗ , P ∗∗ be as in the proof of (i) =⇒ (iv). By [Kec95, 16.1] the set C of all z for which Pz∗∗ is comeager is Borel, so Q(z, y) ⇐⇒ [C(z) =⇒ P ∗∗ (z, y)] is Borel with comeager sections and S(x) = z =⇒ Px = Qz . If moreover P has Gδ sections, we instead let A be the set of all z for which Pz∗∗ is comeager and Gδ , which is Π11 by [Kec95, 35.47]. Then S(X) ⊆ A is Σ11 , so by the Lusin separation theorem there is a Borel set S(X) ⊆ C ⊆ A. We then define Q as above, so that Q moreover has Gδ sections. The result then follows by [Kec95, 35.43]. (D) Theorems 3.1.1 to 3.1.3 are effective, meaning that whenever P is (lightface) ∆11 and satisfies the hypotheses of one of these theorems, then P admits a ∆11 uniformization (cf. [Mos09, 4F.16, 4F.20] and the discussion afterwards). Similarly, [HKL90] implies that if E is smooth and ∆11 then it has a ∆11 reduction to ∆(2N ). The proof of Theorem 3.1.5 therefore gives the following effective refinement: Theorem 3.2.4. Let E be a smooth ∆11 equivalence relation on NN and P ⊆ NN × NN be ∆11 and E-invariant. Then P admits a ∆11 E-invariant uniformization whenever one of the following holds: (i) P has µ-positive sections, for some ∆11 probability measure µ on NN ; (ii) P has non-meager sections; (iii) P has non-empty Kσ sections; (iv) P has non-empty countable sections. 85 In (i) above, we identify probability Borel measures on NN with points in [0, 1]N [Kec95, 17.7]. <N It is also interesting to consider whether the converse holds. For example, let E be a ∆11 equivalence relation on NN , and suppose that for every ∆11 E-invariant set P ⊆ NN ×NN which satisfies one of (i)-(iv) above, P admits a ∆11 E-invariant uniformization. Must it be the case that E is smooth? If we replace ∆11 by Borel, then E must indeed be smooth by Theorem 3.1.5. However, to prove this we use the fact that every non-smooth ∆11 equivalence relation embeds E0 [HKL90], and this is not effective: There are non-smooth ∆11 equivalence relations on NN which do not admit ∆11 embeddings of E0 . Restricting our attention to those P which have countable sections, it turns out that the converse to Theorem 3.2.4 is false. In fact, using the theory of turbulence, one can construct the following very strong counterexample: Theorem 3.2.5. There is a Π01 set N ⊆ NN and a ∆11 equivalence relation E on N which is not smooth, and such that every ∆11 E-invariant set P ⊆ N × NN with non-empty countable sections is invariant, meaning that Px = Px0 for all x, x0 ∈ N . Corollary 3.2.6. There is a ∆11 equivalence relation F on NN which is not smooth, and such that every ∆11 F -invariant set P ⊆ NN × NN with non-empty countable sections admits a ∆11 F -invariant uniformization. Proof. Let N, E be as in Theorem 3.2.5 and define xF x0 ⇐⇒ (x = x0 ) ∨ (x, x0 ∈ N & xEx0 ). Suppose now that P ⊆ NN × NN were ∆11 and F -invariant. Let y ∈ A ⇐⇒ ∃x ∈ N (P (x, y)) ⇐⇒ ∀x ∈ N (P (x, y)). Then A is countable and ∆11 , and Px = A for all x ∈ N . In particular, A contains a ∆11 point, say y0 . By the effective Lusin-Novikov theorem, there is a ∆11 uniformization f of P . Letting g(x) = f (x) for x ∈ / N , and g(x) = y0 otherwise, gives a ∆11 F -invariant uniformization of P . Proof of Theorem 3.2.5. Consider the group RN and the translation action of `1 ⊆ RN on RN , which is turbulent by [Kec03, Section 10(ii)]. Let F be the induced equivalence relation, which is clearly ∆11 . 86 Let F ⊆ N be the Π11 set of codes for the ∆11 functions from RN to NN , and for n ∈ F let fn be the function that it codes. Let also H ⊆ F be the Π11 set of those n for which N fn is a homomorphism of F into ENctble . Finally, let D ⊆ N denote the usual Π11 set of codes for the ∆11 subsets of RN , and for n ∈ D let Dn be the set that it codes. By the proof of [Kec03, Theorem 12.5(i) =⇒ (ii)], for each n ∈ H there is ∆11 comeager N F -invariant set Cn ⊆ RN which fn maps to a single ENctble -class. Moreover, there is a computable map n 7→ n∗ such that if n ∈ H then n∗ ∈ D and Cn = Dn∗ . Put C = T n∈H Cn ⊆ R N . Then C is comeager, F -invariant and Σ11 , since a ∈ C ⇐⇒ ∀n(n ∈ H =⇒ a ∈ Dn∗ ). Moreover, for each ∆11 homomorphism f of F to ENctble , f C maps into a single N ENctble -class. N Let now N ⊆ NN be Π01 and c : N → RN be a ∆11 map such that c(N ) = C. Define the ∆11 equivalence relation E on N by xEx0 ⇐⇒ c(x)F c(x0 ). We will show that this E works. Let P ⊆ N ×NN be E-invariant with non-empty countable sections. Define Q ⊆ C ×NN by (a, y) ∈ Q ⇐⇒ a ∈ C & ∃x ∈ N (c(x) = a & P (x, y)) ⇐⇒ a ∈ C & ∀x ∈ N (c(x) = a =⇒ P (x, y)). Note that Q is F -invariant. Moreover, Q is ∆11 on the Σ11 set C × NN , i.e., it is the intersection of C × NN with a Σ11 set in RN × NN as well as with a Π11 set in RN × NN . By Σ11 separation, there is a ∆11 set R ⊆ RN × NN such that R ∩ (C × NN ) = Q. Let now C ∗ ⊆ RN be defined by a ∈ C ∗ ⇐⇒ ∀a0 [aF a0 =⇒ Ra = Ra0 & Ra is countable and non-empty]. Then C ∗ is Π11 , F -invariant and contains C, so there is a ∆11 set B which is F -invariant and such that C ⊆ B ⊆ C ∗ . Finally, define S ⊆ RN × NN by (a, y) ∈ S ⇐⇒ [a ∈ B & R(a, y)] ∨ [a ∈ / B & y = y0 ] 87 for some fixed ∆11 point y0 in NN . Then S is ∆11 , F -invariant, and has non-empty countable sections. Let s : RN → (NN )N be a ∆11 homomorphism of F to ENctble for which Sa = {s(a)n } for all a ∈ RN , which exists by the effective Lusin-Novikov theorem. By the definition N of C, we have that sC maps into a single ENctble -class. Let A be the corresponding countable set. Then for a ∈ C and any x ∈ N with c(x) = a, N A = Sa = Ra = Qa = Px , so Px = A for all x ∈ N . It remains to check that E is not smooth. To see this, note that F C has at least two classes (as every F -class is meager), and hence so does E. If E were smooth, then there would be a ∆11 map f : N → NN for which xEx0 ⇐⇒ f (x) = f (x0 ). But then graph(f ) would be ∆11 , E-invariant, have non-empty countable sections, and satisfy Px 6= Px0 for some x, x0 ∈ N , a contradiction. Problem 3.2.7. Is there a ∆11 equivalence relation E on NN which is not smooth, and such that all ∆11 E-invariant sets P ⊆ NN × NN satisfying one of (i)-(iii) in Theorem 3.2.4 admit a ∆11 E-invariant uniformization? Finally, we remark that if E is a ∆11 equivalence relation which is not smooth, then there is a continuous embedding of E0 into E which is ∆11 (O). In particular, the converse of Theorem 3.2.4 holds if we consider all such P ∈ ∆11 (O). 3.3 Proofs of Theorems 3.1.6 and 3.1.7 (A) We first prove Theorem 3.1.7. Let F (Y ) denote the Effros Borel space of closed subsets of Y (cf. [Kec95, 12.C]). Suppose Px ∈ Fσ , for all x ∈ X, and that there is an E-invariant Borel map x 7→ Fx ∈ F (Y ) such that Px is non-meager in Fx for all x ∈ X. By [Kec95, 12.13], there is a sequence of E-invariant Borel functions yn : X → Y such that {yn (x)} is dense in Fx for all x ∈ X. Since Px is non-meager and Fσ in Fx , Px contains an open set in Fx , and in particular contains some yn (x). Thus the map taking x to the least yn (x) such that P (x, yn (x)) is an E-invariant Borel uniformization of P . It remains only to show that in each of the cases (i), (ii), (iii), such an assignment x 7→ Fx exists. In (ii), we can take Fx = Y . 88 Consider case (i), that there is a Borel assignment x 7→ µx of probability Borel measures on Y such that Px ∈ ∆02 and µx (Px ) > 0, for all x ∈ X. Let νx denote the probability Borel measure µx restricted to Px , i.e., νx (A) = µx (A ∩ Px )/µx (Px ), and define Fx to be the support of νx , i.e., the smallest νx -conull closed set in Y . Since Fx is the support of νx , any open set in Fx is νx -positive, and therefore any νx -null Fσ set in Fx is meager. Now Px is Gδ and νx -conull in Fx , so Px is comeager in Fx , for all x ∈ X. Thus it remains only to show that the map x 7→ Fx is Borel. To see this, we observe that Fx ∩ U 6= ∅ ⇐⇒ νx (U ) > 0 ⇐⇒ µx (U ∩ Px ) > 0 is Borel, by [Kec95, 17.25]. Finally, consider case (iii), that Px ∈ Gδ and Px is non-empty and Kσ for all x ∈ X. Let Fx be the closure of Px in Y . Then Px is dense Gδ in Fx , so it remains to check that x 7→ Fx is Borel. Indeed, Fx ∩ U 6= ∅ ⇐⇒ Px ∩ U 6= ∅, and this is Borel by the Arsenin-Kunugui theorem [Kec95, 18.18], as Px ∩ U is Kσ for all x ∈ X. (B) We now prove Theorem 3.1.6. Let X = [N]ℵ0 denote the space of infinite subsets of N. By identifying subsets of N with their characteristic functions, we can view X as an E0 -invariant Gδ subspace of 2N . Note that this is a dense Gδ in 2N , and it is µ-conull, where µ is the uniform product measure on 2N . We let E denote the equivalence relation E0 restricted to X. Let Y = 2N , and define P ⊆ X × Y by P (A, B) ⇐⇒ |A \ B| = |A ∩ B| = ℵ0 . Then P is Gδ and E-invariant, and Px is comeager for all x ∈ X. By the Borel-Cantelli lemma, one easily sees that µ(Px ) = 1 for all x ∈ X. We claim that P does not admit an E-invariant Borel uniformization. Indeed, suppose such a uniformization f : X → Y existed. By [Kec95, 19.19], there is some A ∈ X such that f [A]ℵ0 is continuous, where [A]ℵ0 denotes the space of infinite subsets of A. Since E-classes are dense in [A]ℵ0 , f [A]ℵ0 is constant, say with value B. Then f (A) = B, so P (A, B) and A ∩ B is infinite. But then A ∩ B ∈ [A]ℵ0 , so f (A ∩ B) = B. But (A ∩ B) \ B is not infinite, so ¬P (A ∩ B, B), a contradiction. 89 Remark 3.3.1. Using the same Ramsey-theoretic arguments, one can show that the following examples also do not admit E-invariant uniformizations: 1. Let Y be the space of graphs on N and set Q(A, G) iff for all finite disjoint sets x, y ⊆ N there is some a ∈ A which is adjacent (in G) to every element of x and no element of y, i.e., A contains witnesses that G is the random graph. 2. Let Y = [N]ℵ0 , and for B ∈ Y let fB : N → N denote its increasing enumeration. Then take R(A, B) iff fB (A) contains infinitely many even and infinitely many odd elements. As with P above, Q, R both have µ-conull dense Gδ sections. 3.4 3.4.1 Dichotomies and anti-dichotomies Proof of Theorem 3.1.8 Here we derive Miller’s dichotomy Theorem 3.1.8 for sets with countable sections, from Miller’s (G0 , H0 ) dichotomy [Mil12] and Lecomte’s ℵ0 -dimensional hypergraph dichotomy [Lec09]. We begin by noting the following equivalent formulations of the second alternative in Theorem 3.1.8. Proposition 3.4.1. Let X, Y be Polish spaces, E a Borel equivalence relation on X and P ⊆ X × Y an E-invariant Borel relation with countable non-empty sections. Then the following are equivalent: (2) There is a continuous embedding πX : 2N × N → X of E0 × IN into E and a continuous injection πY : 2N × N → Y such that for all x, x0 ∈ 2N × N, ¬(x E0 × IN x0 ) =⇒ PπX (x) ∩ PπX (x0 ) = ∅ and PπX (x) = πY ([x]E0 ×IN ). (3) There is a continuous embedding πX : 2N → X of E0 into E and a continuous injection πY : 2N → Y such that for all x, x0 ∈ 2N , ¬(x E0 x0 ) =⇒ PπX (x) ∩ PπX (x0 ) = ∅ and πY (x) ∈ PπX (x) . 90 (4) There is a continuous embedding πX : 2N → X of E0 into E such that for all x, x0 ∈ 2N , ¬(x E0 x0 ) =⇒ PπX (x) ∩ PπX (x0 ) = ∅. Proof. Clearly (2) =⇒ (3) =⇒ (4). Assume now that (4) holds, and is witnessed by πX . Let g be a uniformization of P and πY = g ◦ πX . Since πY is countable-to-one, by the Lusin-Novikov theorem there is a Borel non-meager set B ⊆ 2N on which πY is injective. We then recursively construct a continuous embedding of E0 into E0 B, and compose this with πX , πY to get maps witnessing (3). Now suppose (3) holds, and is witnessed by πX , πY . Let h be a continuous embedding of E0 × IN into E0 , and let π̃X = πX ◦ h. Let F be the equivalence relation on Y defined by yF y 0 iff y = y 0 or there is some x ∈ 2N × N such that P (π̃X (x), y) and P (π̃X (x), y 0 ). If y 6= y 0 , then the set of x witnessing that yF y 0 is a single E0 × IN -class, so by Lusin-Novikov F is Borel. Thus, πY ◦ h is an embedding of E0 × IN into the countable Borel equivalence relation F , and by compressibility we can turn this into an invariant Borel embedding π̃Y . Now π̃X , π̃Y would be witnesses to (2), except that π̃Y is not necessarily continuous. However, π̃Y is continuous when restricted to an E0 × IN -invariant comeager Borel set C, so it suffices to find a continuous invariant embedding of E0 × IN into (E0 × IN )C. One gets such an embedding by applying [Mild, Proposition 1.4] to the relation xRx0 iff x(E0 × IN )x0 or x ∈ / C or x0 ∈ / C. Remark 3.4.2. From the proof of (3) =⇒ (2), one sees that if E is a countable Borel equivalence relation then actually one can strengthen (2) so that πX is a continuous invariant embedding of E0 × IN into E, i.e., a continuous embedding such that additionally πX ([x]E0 ×IN ) = [πX (x)]E , for all x ∈ 2N × IN . The next two results will be used in the proof of Theorem 3.1.8. Theorem 3.4.3 (Theorem 3.1.9). Let F be a smooth Borel equivalence relation on a Polish space X, Y be a Polish space, and P ⊆ X × Y be a Borel set with countable sections. Suppose that \ Px 6= ∅ x∈C for every F -class C. Then P admits a Borel F -invariant uniformization. 91 Proof. Let Z be a Polish space and S : X → Z be a Borel map such that xF x0 ⇐⇒ S(x) = S(x0 ). Define P ∗ ⊆ Z × Y by P ∗ (z, y) ⇐⇒ ∀x(S(x) = z =⇒ P (x, y)). Note that P ∗ is Π11 , and that if S(x) = z then Pz∗ = \ Px0 xF x0 is non-empty and countable. By Lusin-Novikov, fix a sequence gn of Borel maps gn : X → Y such that P = S ∗ 1 n graph(gn ). Define Q(x, n) ⇐⇒ P (S(x), gn (x)). Then Q is Π1 , so by the number uniformization property [Kec95, 35.1] we can fix a Borel map h uniformizing Q. Let now A(z, y) ⇐⇒ ∃x(S(x) = z & y = gh(x) (x)). Then A ⊆ P ∗ is Σ11 , so by the Lusin separation theorem there is a Borel set A ⊆ P ∗∗ ⊆ P ∗ . By [Kec95, 18.9], the set C = {z | Pz∗∗ is countable} is Π11 , and it contains S(X), so by the Lusin separation theorem again there is some Borel set S(X) ⊆ B ⊆ C. By Lusin-Novikov, there is a Borel uniformization f of R(z, y) ⇐⇒ B(z) & P ∗∗ (z, y). Then f ◦ S is an F -invariant Borel uniformization of P . Proposition 3.4.4. Let E be an analytic equivalence relation on a Polish space X, F ⊇ E be a smooth Borel equivalence relation on X, Y be a Polish space, and P ⊆ X × Y be a Borel E-invariant set with countable sections. Suppose that xF x0 =⇒ Px ∩ Px0 6= ∅ for all x, x0 ∈ X. Then there is a smooth equivalence relation E ⊆ F 0 ⊆ F such that \ Px 6= ∅ x∈C for every F 0 -class C. Proof. Let G ⊆ X N be the ℵ0 -dimensional hypergraph of F -equivalent sequences xn T such that n Pxn = ∅. By Lusin-Novikov, G is Borel. We claim that G has a countable Borel colouring. By [Lec09, Lemma 2.1 and Theorem 1.6], it suffices to show that G has a countable σ(Σ11 )-colouring. Let S 92 be a σ(Σ11 )-measurable selector for F and gn be a sequence of Borel functions such S that P = n graph(gn ). Then the function f (x) assigning to x the least n such that P (x, gn (S(x))) is such a colouring. (In fact, x 7→ gf (x) (S(x)) is a σ(Σ11 )-measurable F -invariant uniformization of P .) If A is G-independent, then so is [A]E . Thus, by repeated application of the first reflection theorem, any G-independent analytic set is contained in an E-invariant Gindependent Borel set. We may therefore fix a countable cover Bn of X by E-invariant G-independent Borel sets. Define xF 0 x0 ⇐⇒ xF x0 & ∀n(x ∈ Bn ⇐⇒ x0 ∈ Bn ). Then F 0 is a smooth Borel equivalence relation and E ⊆ F 0 ⊆ F . Fix x = x0 ∈ X, in order to show that \ Px0 6= ∅. xF 0 x0 Fix an enumeration yn , n ≥ 1 of Px , and suppose for the sake of contradiction that this intersection is empty. Then for each n, there is some xn F 0 x with yn ∈ / Pxn . Also, x ∈ Bk for some k. But then xn ∈ Bk for all k, so Bk is not G-independent, a contradiction. Proof of Theorem 3.1.8. Clearly the two cases are mutually exclusive. To see that at least one of them holds, define the graph G on X by xGx0 ⇐⇒ Px ∩ Px0 = ∅. By Lusin-Novikov, this is a Borel graph. We now apply the (G0 , H0 ) dichotomy [Mil12, Theorem 25] to (G, E), and consider the two cases. Case 1: There is a countable Borel colouring of G ∩ F , where F ⊇ E is smooth. Let A be Borel and (G ∩ F )-independent. By repeated applications of the first reflection theorem, we may assume that A is E-invariant. We can therefore refine F to a smooth equivalence relation F 0 ⊇ E such that xF 0 x0 =⇒ Px ∩ Px0 6= ∅. The result now follows from Theorem 3.4.3 and Proposition 3.4.4. Case 2: Let f be a continuous homorphism from (G0 , H0 ) to (G, E). It suffices to show that (4) holds in Proposition 3.4.1. To see this, consider F = (f × f )−1 (E), R = (f × f )−1 (G). Then H0 ⊆ F and each F -section is G0 -independent, hence meager, so F is meager. We claim R is comeager. To see this, fix x ∈ 2N and consider Rxc = {x0 : Pf (x) ∩ Pf (x0 ) 6= ∅}. Fix an enumeration yn of Pf (x) , and let An = {x0 : S yn ∈ Pf (x0 ) }. Then each An is G0 -independent, hence meager, and Rxc = n An . Thus R has comeager sections, and by Kuratowski-Ulam R is comeager. One can now recursively construct a continuous homomorphism g from ((∆2N )c , Ec0 , E0 ) to ((f × f )−1 (∆X )c , R, E0 ), see e.g. [Milb, Proposition 11]. Then f ◦ g satisfies (4). 93 3.4.2 An ℵ0 -dimensional (G0 , H0 ) dichotomy In this section we state and prove an ℵ0 -dimensional analogue of Miller’s (G0 , H0 ) dichotomy [Mil12, Theorem 25]. This dichotomy generalizes Lecomte’s ℵ0 -dimensional G0 dichotomy [Lec09] (see also [Mil11]) in the same way that Miller’s (G0 , H0 ) dichotomy generalizes the G0 dichotomy [KST99]. We then state an effective analogue of this theorem, and indicate the changes that must be made to prove it. (A) Fix a strictly increasing sequence α ∈ NN and dense sets S ⊆ n N2n , T ⊆ S 2n+1 × N2n+1 , i.e., sets such that for all u ∈ N<N there is some s ∈ S with s ⊆ u, nN and for all (u, v) ∈ N<N × N<N there is some t = (t0 , t1 ) ∈ T such that t0 ⊆ u, t1 ⊆ v. S Let Xα = {x ∈ NN : ∀n∃m ≥ n(xm ∈ α(m)m )}. Note that Xα is dense Gδ in NN . Define the Borel ℵ0 -dimensional directed hypergraph Gω0 on Xα by Gω0 ((xn )) ⇐⇒ ∃s ∈ S∃z ∈ NN ∀n(xn = s_ n_ z), and the Borel directed graph Hω0 on Xα by _ _ _ xHω0 y ⇐⇒ ∃(t0 , t1 ) ∈ T ∃z ∈ NN (x = t_ 0 0 z & y = t1 1 z). We say A ⊆ Xα is Gω0 -independent if x ∈ AN =⇒ ¬Gω0 (x). Proposition 3.4.5 ([Lec09, Lemma 2.1]). Let A ⊆ Xα be Baire measurable and Gω0 -independent. Then A is meager. Proof. Suppose A is non-meager, and fix an open set Ns = {x ∈ NN : s ⊆ x} in which A is comeager. By density of S, we may assume wlog that s ∈ S. For each n, the T set An = {x ∈ NN : s_ n_ x ∈ A} is comeager, so there is some x ∈ n An . But then xn = s_ n_ x ∈ A, and Gω0 ((xn )), so A is not Gω0 -independent. Let R be a quasi-order on a Polish space X. We let ≡R denote the equivalence relation x ≡R y ⇐⇒ xRy & yRx. We say R is lexicographically reducible if there is a Borel reduction of R to the lexicographic order ≤lex on 2α , for some α < ω1 . If A ⊆ X, we let [A]R = {y : ∃x ∈ A(xRy)}, [A]R = {y : ∃x ∈ A(yRx)}, and say A is closed upwards (resp. downward) for R if A = [A]R (resp. A = [A]R ). If A, B ⊆ X, we say (A, B) is R-independent if A × B ∩ R = ∅. Proposition 3.4.6 (Ess. [Milc, Proposition 5]). Let A ⊆ Xα be Baire measurable and ≡Hω0 -invariant. Then A is either meager or comeager. 94 Proof. Suppose A is non-meager, and fix an open set Nu in which A is comeager. We show that A is non-meager in Nv for all v ∈ N<N . By density of T , it suffices to show this assuming that (u, v) ∈ T . The set A0 = {x ∈ NN : u_ 0_ x ∈ A} is comeager, and x ∈ A0 =⇒ v _ 1_ x ∈ A, so A is comeager in Nv_ 1 . Proposition 3.4.7 ([Milc, Proposition 1]). Let R be an analytic quasi-order on a Polish space X and A0 , A1 ⊆ X be analytic such that (A0 , A1 ) is R-independent. Then there are Borel sets Ai ⊆ Bi such that (B0 , B1 ) is R-independent, B0 is closed upwards for R, and B1 is closed downwards for R. Proof. Note that ([A0 ]R , [A1 ]R ) is R-independent, and these sets are analytic. By the first reflection theorem, we can recursively construct a sequence of Borel sets Bni such 0 1 that Ai ⊆ B0i , [Bn0 ]R ⊆ Bn+1 , [Bn1 ]R ⊆ Bn+1 , and (Bn0 , Bn1 ) are R-independent. Take S Bi = n Bni . Let F be an equivalence relation on X and G be an ℵ0 -dimensional directed hypergraph on X. We call A ⊆ X F -locally G-independent if there is no sequence xn ∈ A of pairwise F -equivalent points with G((xn )), and we call c : X → Y an F -local colouring of G if c−1 (y) is F -locally G-independent for all y ∈ Y . Theorem 3.4.8. Let G be an analytic ℵ0 -dimensional directed hypergraph on a Polish space X, and R an analytic quasi-order on X. Then exactly one of the following holds: (1) There is a lexicographically reducible quasi-order R0 on X such that R ⊆ R0 and there is a countable Borel ≡R0 -local colouring of G. (2) There is a continuous homomorphism from (Gω0 , Hω0 ) to (G, R). Proof. To see these are mutually exclusive, it suffices to show that there is no smooth equivalence relation F ⊇ ≡H0ω such that there is a countable Borel F -local colouring c : Xα → N of Gω0 . Arguing by contradiction, suppose such F, c existed. By Proposition 3.4.6, we can fix n ∈ N and a single F -class C such that A = c−1 (n) ∩ C is non-meager. But then by Proposition 3.4.5, A is not Gω0 -independent, a contradiction. We now show that at least one of these alternatives hold. Fix continuous maps πG , πR : NN → X such that G = πG (NN ), R = πR (NN ). Let d denote the usual metric on NN , and dX be a complete metric compatible with the Polish topology on X. 95 Let V be a set, H0 be an ℵ0 -dimensional directed hypergraph on V with edge set E0 , and H1 be a directed graph on V with edge set E1 . A copy of (H0 , H1 ) in (G, R) is a triple ϕ = (ϕX , ϕG , ϕR ) where ϕX : V → X, ϕG : E0 → NN , ϕR : E1 → NN , such that e = (vn ) ∈ E0 =⇒ ϕG (e) = (ϕX (vn ))n∈N , and e = (v, u) ∈ E1 =⇒ ϕR (e) = (ϕX (v), ϕX (u)). Let Hom(H0 , H1 ; G, R) denote the set of all copies of (H0 , H1 ) in (G, R). Note that if V, E0 , E1 are countable, then Hom(H0 , H1 ; G, R) ⊆ X V × (NN )E0 × (NN )E1 is closed, and hence Polish. Suppose now we have H0 , H1 as above, with V, E0 , E1 countable, and consider H ⊆ Hom(H0 , H1 ; G, R). Let H(v) = {ϕX (v) : ϕ ∈ H} for v ∈ V , and note that H(v) is analytic whenever H is analytic. Define H(e) ⊆ NN similarly for e ∈ E0 ∪ E1 . Now call a set H tiny if it is Borel and there is a lexicographically reducible quasi-order R0 on X such that R ⊆ R0 and one of the following holds: (1) H(v) is ≡R0 -locally G-independent for some v ∈ V . (2) ∀ϕ ∈ H∃u, v ∈ V (ϕX (u) 6≡R0 ϕX (v)). In this case, we call R0 a witness that H is tiny, and say H is tiny of type 1 (resp. 2) if H satisfies (1) (resp. (2)). Finally, we say H is small if it is in the σ-ideal generated by the tiny sets, and otherwise we call H large. Finally, fix H0 , H1 as above with V, E0 , E1 countable. For v ∈ V , we define the ℵ0 -dimensional directed hypergraph ⊕v H0 and the directed graph ⊕v H1 by taking a countable disjoint union of H0 (resp. H1 ), on vertex set V × N, and adding the edge (v _ n)n∈N to ⊕v H0 . Similarly, for u, v ∈ V , we define the ℵ0 -dimensional directed hypergraph H0 u +v H0 and the directed graph H1 u +v H1 by taking a countable disjoint union of H0 (resp. H1 ), on vertex set V × N, and adding the edge (u_ 0, v _ 1) to H1 u +v H1 . Note that there are natural continuous projection maps Hom(⊕v H0 , ⊕v H1 ; G, R) → Hom(H0 , H1 ; G, R) and Hom(H0 u +v H0 , H1 u +v H1 ; G, R) → Hom(H0 , H1 ; G, R), 96 for all n ∈ N, taking ϕ to its restriction ϕn to V × {n}. If H ⊆ Hom(H0 , H1 ; G, R), we let ⊕v H = {ϕ ∈ Hom(⊕v H0 , ⊕v H1 ; G, R) : ∀n(ϕn ∈ H)}, H u +v H = {ϕ ∈ Hom(H0 u +v H0 , H1 u +v H1 ; G, R) : ∀n(ϕn ∈ H)}. Claim 3.4.9. If Hom(·, ·; G, R) is small, then there is a lexicographically reducible quasi-order R0 on X such that R ⊆ R0 and there is a countable Borel ≡R0 -local colouring of G. Proof. Note that Hom(·, ·; G, R) can be naturally identified with X, so that our assumption implies that X can be covered by countably-many Borel sets An such that for each n, there is a lexicographically reducible quasi-order Rn such that R ⊆ Rn and An is ≡Rn -locally G-independent. Let fn : X → 2αn be a Borel reduction of Rn to the lexicographic ordering on P 2αn , αn < ω1 . Let α = n αn , and consider the map f : X → 2α , f (x) = f0 (x)_ f1 (x)_ f2 (n)_ · · · . Then f is Borel, so xR0 y ⇐⇒ f (x) ≤lex f (y) is a lexiT cographically reducible quasi-order containing R. Note also that ≡R0 = n ≡Rn . It follows that the map taking x to the least n for which x ∈ An is a countable Borel ≡R0 -local colouring of G. Claim 3.4.10. Let H0 , H1 be as above with V, E0 , E1 countable, F ⊆ V ∪ E0 ∪ E1 be finite, ε > 0, and H ⊆ Hom(H0 , H1 ; G, R) be large and Borel. Then there is a large Borel set H0 ⊆ H for which diamdX (H0 (v)) < ε for all v ∈ F ∩ V and diamd (H0 (e)) < ε for all e ∈ F ∩ (E0 ∪ E1 ). Proof. This follows from the fact that we can cover X, NN with countably many sets of small diameter, and the small sets form a σ-ideal. Claim 3.4.11. Let H0 , H1 be as above with V, E0 , E1 countable, and suppose H ⊆ Hom(H0 , H1 ; G, R) is Borel and large. Then ⊕v H, H u +v H are Borel and large. Proof. That these sets are Borel is clear. Now suppose ⊕v H is small and write S ⊕v H = i∈2,n∈N Fni , with Fni tiny of type i and witness Rni . Arguing as in the proof of Claim 3.4.9, we may assume that Rni = R0 for a single quasi-order R0 . Let vn ∈ V be such that Fn0 (vn ) is ≡R0 -locally G-independent. By the first reflection theorem, 97 we may fix Borel sets Fn0 (vn ) ⊆ An which are ≡R0 -locally G-independent. Define Hn = {ϕ ∈ H : ϕX (vn ) ∈ An }, and let ! 0 H = H \ {ϕ ∈ H : ∃u, v ∈ V (ϕX (u) 6≡R0 ϕX (v))} ∪ [ Hn . n We claim H0 is tiny, which implies that H is small. Clearly H0 is Borel, and we claim H0 (v) is ≡R0 -locally G-independent. Indeed, if ϕn ∈ H0 and G(((ϕn )X (v))n∈N ), then there is some ϕ ∈ ⊕v H with ϕn = ϕn for all n. But then ϕ ∈ Fn1 for some n, so there are u, w ∈ V ×N such that ϕX (u) 6≡R0 ϕX (w). Since ϕn ∈ H0 for all n, we may assume that u = v _ i, w = v _ j for some i 6= j. But then ϕiX (v) = (ϕi )X (v) 6≡R0 (ϕj )X (v) = ϕjX (v). Next suppose H u +v H is small and write H u +v H = i∈2,n∈N Fni , with Fni tiny of type i and witness Rni . As before, we may assume Rni = R0 , and we define Hn , H0 in the same way, so that it suffices to show that H0 is tiny of type 2. S Let ϕi ∈ H0 , i ∈ 2, and suppose (ϕ0 )X (u)R(ϕ1 )X (v). Then there is some ϕ ∈ H u +v H with ϕ0 = ϕ0 and ϕi = ϕ1 for i > 0. As before, we find that we must have ϕX (u_ 0) 6≡R0 ϕX (v _ 1), so that (ϕ0 )X (u) 6≡R0 (ϕ1 )X (v). Thus, (H0 (u), H0 (v)) is (R ∩ ≡R0 )-independent, and by Proposition 3.4.7 we can find Borel sets H0 (u) ⊆ A, H0 (v) ⊆ B such that A is closed upwards for R ∩ ≡R0 , B is closed downwards for R ∩ ≡R0 , and (A, B) is (R ∩ ≡R0 )-independent. Then xQy ⇐⇒ xR0 y & (x ≡R0 y & x ∈ A =⇒ y ∈ A) is a lexicographically reducible quasi-order containing R, and H0 is tiny of type 2 with witness Q. If Hom(·, ·; G, R) is small, then by Claim 3.4.9 we are done. Suppose now that Hom(·, ·; G, R) is large. We define a sequence Gn of ℵ0 -dimensional directed graphs on Nn and a sequence Hn of directed graphs on Nn as follows: Gn (xi ) ⇐⇒ ∃k < n ∃s ∈ (S ∩ Nk ) ∃u ∈ Nn−k−1 ∀i(xi = s_ i_ u), xHn y ⇐⇒ ∃k < n ∃(t0 , t1 ) ∈ (T ∩ Nk × Nk ) _ _ _ ∃u ∈ Nn−k−1 (x = t_ 0 0 u & y = t1 1 y). Note that if s ∈ S ∩ Nn then Gn+1 = ⊕s Gn and Hn+1 = ⊕s Hn , and if (t0 , t1 ) ∈ T ∩ Nn × Nn then Gn+1 = Gn t0 +t1 Gn and Hn+1 = Hn t0 +t1 Hn . Also, Gω0 ((xi )i∈N ) ⇐⇒ ∃N ∀n ≥ N (Gn ((xi n)i∈N )) 98 and xHω0 y ⇐⇒ ∃N ∀n ≥ N (xn Hn yn), and Gn , Hn have countably many vertices and edges. By Claims 3.4.10 and 3.4.11, we can recursively construct a sequence of large Borel sets Hn ⊆ Hom(Gn , Hn ; G, R) such that Hn+1 ⊆ Hn ⊕s Hn for s ∈ S ∩ Nn , Hn+1 ⊆ Hn t0 +t1 Hn for (t0 , t1 ) ∈ T ∩ Nn × Nn , diamdX (Hn (x)) < 2−n for all x ∈ α(n)n , and diamd (H(e)) < 2−n for all e ∈ Gn ∪ Hn with e0 ∈ α(n)n , where e0 denotes the first T vertex in e. It follows that {f (x)} = n Hn (xn) exists and is well defined for x ∈ Xα , and that this map f : Xα → X is continuous. To see that it is a homomorphism of Gω0 to G, suppose Gω0 ((xi )i∈N ) and let N be sufficiently large that GN ((xi N )i∈N ). T Then {y} = n≥N Hn ((xi n)i∈N ) exists and is well defined, and by continuity we have (f (xi ))i∈N = πG (y) ∈ G. A similar argument shows that f is a homomorphism from Hω0 to R. (B) This dichotomy admits the following effective refinement: Theorem 3.4.12. Let G be a Σ11 ℵ0 -dimensional directed hypergraph on a Polish space X, and R a Σ11 partial order on X. Then exactly one of the following holds: 1. There is a quasi-order R0 on X such that R ⊆ R0 , there is a countable ∆11 ≡R0 -local colouring of G, and there is a ∆11 reduction of R0 to the lexicographic order ≤lex on 2α , for some α < ω1CK . 2. There is a continuous homomorphism from (Gω0 , Hω0 ) to (G, R). To prove this, we make the following modifications to the proof of Theorem 3.4.8. First, we choose πG , πH to be computable (restricting their domains appropriately to Π01 sets). We then replace “Borel” with “∆11 ” and “lexicographically reducible” with “admitting a ∆11 reduction to ≤lex on 2α , for some α < ω1CK ” in the definition of tiny sets. We now have the following: Lemma 3.4.13. Let V be a set, H0 be an ℵ0 -dimensional directed hypergraph on V with edge set E0 and H1 be a directed graph on V with edge set E1 , with V, E0 , E1 countable. Suppose H ⊆ Hom(H0 , H1 ; G, R) is small and ∆11 . Then one can find: (1) a uniformly ∆11 sequence of tiny sets (Fni )i∈2,n∈N covering H, 99 (2) a uniformly ∆11 sequence (Rni )i∈2,n∈N of quasi-orders on N, (3) a uniformly ∆11 sequence of ordinals αni < ω1CK , (4) a uniformly ∆11 sequence (fni )i∈2,n∈N of maps fni : NN → 2αn , and i (5) a uniformly ∆11 sequence vn ∈ V , such that the sets Fni are pairwise disjoint, R ⊆ Rni for all i, n, each fni is a reduction i of Rni to ≤lex on 2αn , Fn0 (vn ) are ≡Rn0 -locally G-independent, and ∀ϕ ∈ H∃u, v ∈ V (ϕX (u) 6≡Rn1 ϕX (v)). Proof sketch. Fix a nice coding D 3 n 7→ Dn of the ∆11 sets. The assertion that (F, R0 , α, f, v, i) is a witness that F is tiny of type i is Π11 -on-∆11 . It follows that the relation “ϕ ∈ / H or n ∈ D codes such a tuple with ϕ ∈ F ” is Π11 , and hence by the number uniformization theorem for Π11 there is a ∆11 map g : H → D taking each ϕ ∈ H to such a tuple. The image of H under g is Σ11 , so by the Lusin separation theorem there is a ∆11 set A ⊆ D containing g(H) and such that every element in A codes a tuple as above. One can then fix a ∆11 enumeration of A, which satisfies all of the above conditions except maybe pairwise disjointness of the family Fni , and this can be fixed by a straightforward recursive construction. The effective analogue of Claim 3.4.9 follows immediately. We note that the first reflection theorem is effective enough that the proof of Proposition 3.4.7 is effective as well. Claim 3.4.11 now follows using Lemma 3.4.13. The rest of the proof is identical to that of Theorem 3.4.8. 3.4.3 Proof of Theorem 3.1.8 from the ℵ0 -dimensional (G0 , H0 ) dichotomy (A) Clearly the two cases are mutually exclusive. To see that at least one of them T holds, define the ℵ0 -dimensional hypergraph G on X by G(xn ) ⇐⇒ n Pxn = ∅. By Lusin-Novikov, G is Borel. We now apply Theorem 3.4.8 to (G, E), and consider the two cases. Case 1: There is a lexicographically reducible quasi-order R containing E and a countable Borel ≡R -local colouring of G. Let F = ≡R , so that E ⊆ F and F is smooth. Since P is E-invariant, if A is F -locally G-independent then so is [A]E . It follows that there is a countable Borel E-invariant F -local colouring of G, so that after refining F with this colouring we may assume that X is F -locally G-independent, i.e., T x∈C Px 6= ∅ for every F -class C. Then P admits a Borel F -invariant uniformization by Theorem 3.4.3. 100 Case 2: There is a continuous homomorphism π : Xα → X of (Gω0 , Hω0 ) to (G, E). We will show that (4) holds in Proposition 3.4.1. To see this, consider F = (π × π)−1 (E) and R = (π × π)−1 (R0 ), where xR0 x0 ⇐⇒ Px ∩ Px0 = ∅. Note that R0 is Borel by Lusin-Novikov, and hence so is R. Also, Hω0 ⊆ F and F ∩ R = ∅. We claim that R is comeager. To see this, fix x ∈ Xα and consider Rxc = {x0 ∈ Xα : Pπ(x) ∩ Pπ(x0 ) 6= ∅}. Fix an enumeration yn of Pπ(x) , and let An = {x0 ∈ Xα : yn ∈ Pπ(x0 ) }. Then each An S is Gω0 -independent and hence meager; thus so is Rxc = n An . Thus Rx is comeager for all x ∈ Xα , and by Kuratowski-Ulam R is comeager. One can now recursively construct a continuous homomorphism f : 2ω → Xα from (∆(2ω )c , Ec0 , E0 ) to ((π × π)−1 (∆(X))c , R, F ), see e.g. [Milb, Proposition 11]. Then π ◦ f satisfies (4). (B) We note the following effective version of Theorem 3.1.8: Theorem 3.4.14. Let E be a ∆11 equivalence relation on NN and P ⊆ NN × NN an E-invariant ∆11 relation with countable non-empty sections. Then exactly one of the following holds: 1. There is a ∆11 E-invariant uniformization, 2. There is a continuous embedding πX : 2N × N → X of E0 × IN into E and a continuous injection πY : 2N × N → Y such that for all x, x0 ∈ 2N × N, ¬(x E0 × IN x0 ) =⇒ PπX (x) ∩ PπX (x0 ) = ∅ and PπX (x) = πY ([x]E0 ×IN ). This follows from the above proof, Theorem 3.4.12, and the fact that the proof of Theorem 3.4.3 is effective. 3.4.4 Proof of Theorem 3.1.10 (A) Note first that (1) is equivalent to the existence of a smooth Borel equivalence F ⊇ E for which P is F -invariant, by Theorem 3.2.3. To see that these are mutually exclusive, let F ⊇ E be smooth so that P is F -invariant, and suppose that πX , πY witness (2). Then there is a comeagre E0 -invariant set C 101 that πX maps into a single F -class, so πY (C) is contained in a single P -section, a contradiction. Now define the graph xGx0 ⇐⇒ Px 6= Px0 . This graph is Borel by Lusin-Novikov. Apply the (G0 , H0 ) dichotomy to (G, E). Case 1: There is a smooth F ⊇ E such that G admits a countable Borel F -local colouring. If A is analytic and F -locally G-independent, then so is [A]E , so by repeated applications of the first reflection theorem it is contained in a Borel E-invariant F locally G-independent set. Thus we may assume that G admits a countable Borel E-invariant F -local colouring, and hence by refining F with this colouring we may assume that actually G ∩ F = ∅, i.e., P is F -invariant. Thus (1) holds. Case 2: There is a continuous homomorphism ϕ : 2N → X of (G0 , H0 ) to (G, E). Define R(x, y) ⇐⇒ P (ϕ(x), y), and let Q(x, y) ⇐⇒ R(x, y) & ∀∗ x0 ¬R(x0 , y), where ∀∗ xA(x) means A is comeager for A ⊆ 2N . Let A = proj(Q) and xSx0 ⇐⇒ Qx ∩ Qx0 6= ∅. Then R is Borel with countable sections, and it follows that Q, A, S are Borel as well. Additionally, R, Q are E0 -invariant. We claim that A is comeager and S is meager. Granted this, we can find a continuous homomorphism ψ : 2N → A of (E0 , Ec0 ) to (E00 , S c ) such that ϕ ◦ ψ is injective, where E00 is the smallest equivalence relation containing H0 (cf. [Milb, Proposition 11]). Now the set Q0 (x, y) ⇐⇒ Q(ψ(x), y) has countable sections, so it admits a Borel uniformization g. Since ψ is a homomorphism from Ec0 to S c , g is countable-to-one, so by Lusin-Novikov it is injective and continuous on a non-meager set B. Let τ be a continuous embedding of E0 into E0 B. Then πX = ϕ ◦ ψ ◦ τ, πY = g ◦ τ satisfy (2). Now suppose that A is comeager, in order to show that S is meager. By KuratowskiUlam, it suffices to show that Sx is meager for all x ∈ A. So consider x ∈ A, and let y ∈ Qx be arbitrary. Then y ∈ / Rx0 for comeagerly-many x0 , and so y ∈ / Qx0 for comeagerly-many x0 . Since Qx is countable, it follows that Qx ∩ Qx0 = ∅ for comeagerly-many x0 , and so Sx is meager. It remains to show that A is comeager. To see this, define xBx0 ⇐⇒ Rx ⊆ Rx0 . For T any x, Bx = y∈Rx {x0 : R(x0 , y)}, so if Bx is meager then there is some y ∈ Rx for which {x0 : R(x0 , y)} is not comeager. But this set is Borel and E0 -invariant, so it is meager, and hence y ∈ Qx and x ∈ A. Thus by Kuratowski-Ulam it suffices to show that B is meager. 102 Suppose for the sake of contradiction that B is non-meager. Let C be the set of all x so that Bx is non-meager. Since Bx is E0 -invariant, it must be comeager for all x ∈ C. Moreover C is non-meager and E0 -invariant, and therefore it is comeager. It follows that B is comeager, and hence so is B 0 (x, x0 ) ⇐⇒ B(x, x0 ) & B(x0 , x). In particular, Bx0 = C is comeager for some x. But then x, x0 ∈ C =⇒ Rx = Rx0 ; thus C is G0 -independent, a contradiction. Remark 3.4.15. This proof actually shows that in case (2), we can take πX , πY so that additionally πY (x) ∈ PπX (x0 ) ⇐⇒ xE0 x0 . (B) This proof can also be made effective, by Theorem 3.4.12: Theorem 3.4.16. Let E be a ∆11 equivalence relation on X and P ⊆ NN × NN an E-invariant ∆11 relation with countable non-empty sections. Then exactly one of the following holds: (1) There is a uniformly ∆11 sequence gn : X → Y of E-invariant uniformizations with S P = n graph(gn ), (2) There is a continuous embedding πX : 2N → X of E0 into E and a continuous injection πY : 2N → Y such that for all x ∈ 2N , P (πX (x), πY (x)). 3.4.5 Proofs of Proposition 3.1.11 and Theorem 3.1.12 Let us fix a parametrization of the Borel relations on NN , as in [AK00, Section 5] (see also [Mos09, Section 3H]). This consists of a set D ⊆ 2N and two sets S, P ⊆ (NN )3 such that (i) D is Π11 , S is Σ11 and P is Π11 ; (ii) for d ∈ D, Sd = Pd , and we denote this set by Dd ; (iii) every Borel set in (NN )2 appears as Dd for some d ∈ D; and (iv) if B ⊆ X ×(NN )2 is Borel, X a Polish space, there is a Borel function p : X → 2N so that Bx = Dp(x) for all x ∈ X. Define P = {(d, e) : Dd is an equivalence relation on NN and De is Dd -invariant}, 103 and let P unif denote the set of pairs (d, e) ∈ P for which De admits a Dd -invariant uniformization. More generally, for any set A of properties of sets P ⊆ NN × NN , let PA (resp. PAunif ) denote the set of pairs (d, e) in P (resp. P unif ) such that De unif satisfies all of the properties in A. Let Pctble (resp. Pctble ) denote PA (resp. PAunif ) for A consisting of the property that P has countable sections. We are interested in properties asserting that De , or its sections, are Gδ , Fσ , comeager, non-meager, µ-positive, µ-conull, countable, or Kσ , where µ varies over probability Borel measures on NN . It is straightforward to check, using [Kec95, 16.1, 17.25, 18.9, 35.47], that for all such sets of properties A, PA is Π11 and PAunif is Σ12 . unif By Theorem 3.4.14, we can bound the complexity of Pctble : unif Proposition 3.4.17 (Proposition 3.1.11). The set Pctble is Π11 . unif Proof. By Theorem 3.4.14, (d, e) ∈ Pctble iff (d, e) ∈ Pctble and there exists a ∆11 (d, e) function f which is a Dd -invariant uniformization of De . The assertion that a ∆11 (d, e) function f is a Dd -invariant uniformization of De is Π11 (d, e), so by bounded quantifiunif cation for ∆11 [Mos09, 4D.3], Pctble is Π11 . Recall that a set B in a Polish space X is called Σ12 -complete if it is Σ12 , and for all zero-dimensional Polish spaces Y and Σ12 sets C ⊆ Y there is a continuous function f : Y → X such that C = f −1 (B). Note that by [Sab12, Theorem 2], one could equivalently take f to be Borel in this definition (see also [Paw15]). The following computes the exact complexity of the sets PAunif , when A asserts that De has “large” sections. Theorem 3.4.18 (Theorem 3.1.12). The set PAunif is Σ12 -complete, where A is one of the following sets of properties of P ⊆ NN × NN : 1. P has non-meager sections; 2. P has non-meager Gδ sections; 3. P has non-meager sections and is Gδ ; 4. P has µ-positive sections for some probability Borel measure µ on NN ; 5. P has µ-positive Fσ sections for some probability Borel measure µ on NN ; 6. P has µ-positive sections for some probability Borel measure µ on NN and is Fσ . 104 The same holds for comeager instead of non-meager, and µ-conull instead of µ-positive. In fact, there is a hyperfinite Borel equivalence relation E with code d ∈ D such that for all such A above, the set of e ∈ D such that (d, e) ∈ PAunif is Σ12 -complete. Proof. We will show this first when A asserts that P is Gδ and has comeager sections. Since NN is Borel isomorphic to NN × 2N , we may assume that Dd is instead an equivalence relation on NN × 2N , and that De ⊆ (NN × 2N ) × NN . Let E be the hyperfinite Borel equivalence relation on NN × 2N given by (x, y)E(x0 , y 0 ) ⇐⇒ x = x0 & yE0 y 0 , fix a code d ∈ D for E, and let PAunif (E) denote the set of all e ∈ D so that (d, e) ∈ PAunif . We will show that PAunif (E) is Σ12 -complete. Let now T be a tree on N × N (cf. [Kec95, 2.C]). Each such tree T defines a closed subset [T ] ⊆ NN × NN given by [T ] = {(x, y) ∈ NN × NN : ∀n ((xn, yn) ∈ T )}. We say [T ] admits a full Borel uniformization if there is a Borel map f : NN → NN so that (x, f (x)) ∈ [T ] for all x ∈ NN , and we denote by FBU the set of trees on N × N which admit full Borel uniformizations. By the proof of Theorem 3.1.5, and considering NN as a co-countable set in 2N , there is a Gδ set P ⊆ 2N × NN with comeager sections which is E0 invariant, and so that \ Px = ∅ x∈C whenever C ⊆ 2N is µ-positive, where µ is the uniform product measure on 2N . Given a tree T on N × N, define PT ⊆ (NN × 2N ) × NN by PT (x, y, z) ⇐⇒ P (y, z) ∨ (x, z) ∈ [T ]. Note that PT is Gδ , E-invariant, and has comeager sections. Claim 3.4.19. [T ] admits a full Borel uniformization iff PT admits a Borel E-invariant uniformization. Proof. If f is a full Borel uniformization of [T ], then g(x, y) = f (x) is an E-invariant Borel uniformization of PT . Conversely, suppose g were an E-invariant Borel uniformization of PT . For x ∈ NN , let gx (y) = g(x, y). Then gx : 2N → NN is E0 -invariant, 105 hence constant on a µ-conull set C ⊆ 2N . Since \ Py = ∅, y∈C we cannot have P (y, gx (y)) for all y ∈ C, and so (x, gx (y)) ∈ [T ] for all y ∈ C. Thus f (x) = z ⇐⇒ ∀∗µ y(g(x, y) = z) is a full Borel uniformization of [T ] (cf. [Kec95, 17.26] and the paragraphs following it). By identifying trees on N × N with their characteristic functions, we can view the space of trees as a closed subset of 2N . The set B given by B(T, x, y, z) ⇐⇒ T is a tree and PT (x, y, z) is clearly Borel, so there is a Borel map p such that for each tree T , p(T ) ∈ D and Dp(T ) = PT . It follows by Claim 3.4.19 that FBU = p−1 (PAunif (E)). By [AK00, Lemma 5.3], the set FBU is Σ12 -complete, and hence so is PAunif (E). The cases 1–3 follow from this as well. For 4–6, simply replace P in the above proof with an Fσ set Q ⊆ 2N × NN with µ-conull sections which is E0 -invariant, and so that \ Qx = ∅ x∈C whenever C ⊆ 2N is non-meager, which exists by the proof of Theorem 3.1.5. Remark 3.4.20. We do not know the complexity of PAunif when A asserts that P is Gδ and has comeager µ-conull sections for a probability Borel measure µ. By the proof of Theorem 3.1.6, there is an E0 -invariant Gδ set R ⊆ [N]ℵ0 × NN with comeager µ-conull sections, such that \ Px = ∅ x∈C for all Ramsey-positive sets C ⊆ [N] . One can define PT for a tree T on N × N as in the proof of Theorem 3.1.12, however the “if” direction of our proof of Claim 3.4.19 no longer works (cf. [Sab12]). ℵ0 3.4.6 Proof of Proposition 3.1.14 By [Kec95, 18.17], there is a Gδ set R ⊆ NN × 2N with projNN (R) = NN which does not T admit a Borel uniformization. Write R = n Qn , Qn ⊆ NN × 2N open, and define P by P (n, x, y) ⇐⇒ Qn (x, y). 106 Let (n, x)F (m, x0 ) ⇐⇒ x = x0 . Then F is a smooth countable Borel equivalence relation, P is open, and if C = [(n, x)]F is an F -class then \ Pu = \ n u∈C P(n,x) = \ (Qn )x = Rx 6= ∅. n Suppose now towards a contradiction that g : N × NN → 2N is an F -invariant uniformization of P . Define f : NN → 2N by f (x) = g(0, x). Then f (x) = g(0, x) = T g(n, x) ∈ P(n,x) for all n, so f (x) ∈ n P(n,x) = Rx , a contradiction. 3.5 On Conjecture 3.1.15 Concerning Conjecture 3.1.15, we first note the following analog of Lemma 3.2.1. Lemma 3.5.1. Let E, F be Borel equivalence relations on Polish spaces X, X 0 , resp., such that E ≤B E 0 . If E fails (b) (resp., (c), (d)), so does E 0 . The proof is identical to that of Lemma 3.2.1. Note now that any countable Borel equivalence relation E trivially satisfies (b), (c), and (d), so by Lemma 3.5.1, in Conjecture 3.1.15, (a) implies (b), (c), and (d). To verify then Conjecture 3.1.15, one needs to show that if E is not reducible to countable, then (b), (c), and (d) fail. It is an open problem (see [HK01, end of Section 6]) whether the following holds: Problem 3.5.2. Let E be a Borel equivalence relation which is not reducible to countable. Then one of the following holds: (1) E1 ≤B E, where E1 is the following equivalence relation on (2N )N : xE1 y ⇐⇒ ∃m∀n ≥ m(xn = yn ); (2) There is a Borel equivalence relation F induced by a turbulent continuous action of a Polish group on a Polish space such that F ≤B E; (3) EN0 ≤B E, where EN0 is the following equivalence relation on (2N )N : xEN0 y ⇐⇒ ∀n(xn E0 yn ). It is therefore interesting to show that (b), (c), and (d) fail for E1 , F as in (2) above, and EN0 . Here are some partial results. 107 Proposition 3.5.3. Let E be a Borel equivalence relation which is not reducible to countable but is Borel reducible to a Borel equivalence relation F with Kσ classes. Then E fails (d). In particular, E1 and E2 fail (d), where E2 is the following equivalence relation on 2N : X 1 xE2 y ⇐⇒ < ∞. n:xn 6=yn n + 1 Proof. Suppose E, F live on the Polish spaces X, Y , resp., and let g : X → Y be a Borel reduction of E to F . Define P ⊆ X × X as follows: (x, y) ∈ P ⇐⇒ g(x)F y. Clearly P is E-invariant and has Kσ sections. Suppose then that P admitted a Borel E-invariant countable uniformization f : X → Y N . Then define h : X → X by g(x) = f (x)0 . Then by [Kec25, Proposition 3.7], h shows that E is reducible to countable, a contradiction. Concerning (b) and (c) for E1 , the following is a possible example for their failure. Problem 3.5.4. Let X = (2N )N , Y = 2N and define P ⊆ X × Y as follows: (x, y) ∈ P ⇐⇒ ∃m∀n ≥ m(xn 6= y), so that P is E1 -invariant and each section Px is co-countable, so has µ-measure 1 (for µ the product measure on Y ) and is comeager. Is there a Borel E1 -invariant countable uniformization of P ? One can show the following weaker result, which provides a Borel anti-diagonalization theorem for E1 . Proposition 3.5.5. Let f : (2N )N → 2N be a Borel function such that xE1 y =⇒ f (x) = f (y). Then there is x ∈ (2N )N such that for infinitely many n, f (x) = xn . Thus if X, Y, P are as in Problem 3.5.4, P does not admit a Borel E1 -invariant uniformization. Proof. For any nonempty countable set S ⊆ 2N consider the product space S N with the product topology, where S is taken to be discrete. Denote by E0 (S) the equivalence relation on S N given by xE0 (S)y ⇐⇒ ∃m∀n ≥ m(xn = yn ). This is generically ergodic and for x, y ∈ S N we have that xE0 (S)y =⇒ f (x) = f (y), so there is (unique) 108 xS ∈ 2N such that f (x) = xS , for comeager many x ∈ S N . Clearly xS can be computed in a Borel way given any x ∈ (2N )N with S = {xn : n ∈ N}, i.e., we have a Borel function F : (2N )N → 2N such that {xn : n ∈ N} = {yn : n ∈ N} = S =⇒ F ((xn )) = F ((yn )) = xS . We now use the following Borel anti-diagonalization theorem of H. Friedman, see [Sta85, Theorem 2, page 23]: Theorem 3.5.6 (H. Friedman). Let E be a Borel (even analytic) equivalence relation on a Polish space X. Let F : X N → X be a Borel function such that {[xn ]E : n ∈ N} = {[yn ]E : n ∈ N} =⇒ F ((xn )) E F ((yn )). Then there is x ∈ X N and i ∈ N such that F (x)Exi . Applying this to E being the equality relation on 2N and F as above, we conclude that for some S, we have that xS ∈ S. Then for comeager many x ∈ S N we have that xn = xS , for infinitely many n, and also (x, xS ) ∈ P , a contradiction. In response to a question by Andrew Marks, we note the following version of Proposition 3.5.5 for E1 restricted to injective sequences. Below [2N ]N is the Borel subset of (2N )N consisting of injective sequences and x ≤T y means that x is recursive in y. Proposition 3.5.7. Let g : [2N ]N → 2N be a Borel function such that xE1 y =⇒ g(x) = g(y). Then there is y ∈ [2N ]N such that for all n, g(y) ≤T yn . Proof. Fix a recursive bijection x 7→ hxi from (2N )N to 2N and for each i ∈ N let ī ∈ 2N be the characteristic function of {i}. Then for each x ∈ (2N )N and i ∈ N, put x̄i = hī, xi , xi+1 , . . . i ∈ 2N and x0 = hx̄0 , x̄1 , . . . i ∈ [2N ]N . Note that xE1 y =⇒ x0 E1 y 0 . Finally define f : (2N )N → 2N by f (x) = g(x0 ). Then by Proposition 3.5.5, there is x ∈ (2N )N such that for infinitely many n we have that f (x) = xn . Let y = x0 . If n is such that f (x) = g(y) = xn , then as xn ≤T x̄k = yk , ∀k ≤ n, we have that g(y) ≤T yk , ∀k ≤ n. Since this happens for infinitely many n, we have that g(y) ≤T yn , for all n. 109 We do not know anything about EN0 but if we let Ectble be the equivalence relation N E2ctble (so that EN0 <B Ectble ), we have: Proposition 3.5.8. Ectble fails (b) and (c). Proof. We will prove that Ectble fails (b), the proof that it also fails (c) being similar. Let X = (2N )N , Y = 2N , let µ be the usual product measure on Y and put E = Ectble . Define P ⊆ X × Y by (x, y) ∈ P ⇐⇒ y ∈ / {xn : n ∈ N}. Clearly µ(Px ) = 1 and P is E-invariant. Assume now, towards a contradiction, that there is a Borel function f : X → Y N such that ∀x ∈ X∀n ∈ N((x, f (x)n ) ∈ P ) and x1 Ex2 =⇒ {f (x1 )n : n ∈ N} = {f (x2 )n : n ∈ N}. Then   ∀x ∈ X {f (x)n : n ∈ N} ∩ {xn : n ∈ N} = ∅ . Define F : X N → Y N as follows: Fix a bijection (i, j) 7→ hi, ji from N2 to N and for n ∈ N put n = hn0 , n1 i. Given x ∈ X N , define x0 ∈ X by x0n = (xn0 )n1 . Then let F (x) = f (x0 ). First notice that for x = (xn ), y = (yn ) ∈ X N , {[xn ]E : n ∈ N} = {[yn ]E : n ∈ N} =⇒ x0 Ey 0 =⇒ F (x)EF (y). Thus by Theorem 3.5.6, there is some x ∈ X N and i ∈ N such that F (x)Exi , i.e., f (x0 )Exi or {f (x0 )n : n ∈ N} = {(xi )n : n ∈ N} = {x0hi,ni : n ∈ N}. Thus {f (x0 )n : n ∈ N} ∩ {x0n : n ∈ N} 6= ∅, a contradiction. We do not know if Ectble fails (d). We also do not know anything about equivalence relations induced by turbulent continuous actions of Polish groups on Polish spaces. Finally, we note that by the dichotomy theorem of Hjorth concerning reducibility to countable (see [Hjo05] or [Kec25, Theorem 3.8]), in order to prove Conjecture 3.1.15 for Borel equivalence relations induced by Borel actions of Polish groups, it would be sufficient to prove it for Borel equivalence relations induced by stormy such actions. 110 Chapter 4 INVARIANT UNIFORMIZATION OVER QUOTIENTS Michael S. Wolman 4.1 Introduction Let X, Y be sets and P ⊆ X × Y . For x ∈ X, we let Px = {y ∈ Y : (x, y) ∈ P } denote the section of P above x. When P has non-empty sections, we say a function f : X → Y is a uniformization of P if (x, f (x)) ∈ P for x ∈ X. A (proper) quasi-uniformization of P is a set U ⊆ X × Y such that for all x ∈ X, Ux is a (proper) non-empty finite subset of Px . Let now E, F be Borel equivalence relations on Polish spaces X, Y , i.e., equivalence relations so that E ⊆ X 2 , F ⊆ Y 2 are Borel. We say P ⊆ X/E × Y /F is (weakly) Borel if its lift P̃ = {(x, y) ∈ X × Y : ([x]E , [y]F ) ∈ P } is Borel, and a function f : X/E → Y /F is (strongly) Borel if its graph is Borel. (We note that in [dRM] these notions are referred to as weakly and strongly Borel. In this paper we simply refer to them as Borel, as there is no ambiguity in what we mean in various contexts.) We are interested in characterizing exactly when a Borel set P ⊆ X/E × Y /F with countable sections admits a Borel (proper quasi-)uniformization. For notational convenience, we will always assume that the sections of P are non-empty, though in most cases this is equivalent to the general case (see e.g. [dRM, Theorem 2.12]). When E, F are the equality relations ∆(X), ∆(Y ), Borel uniformizations always exist: Theorem 4.1.1 (Lusin–Novikov). Let X, Y be Polish spaces and P ⊆ X × Y be a Borel set with countable non-empty sections. Then P admits a Borel uniformization. Moreover, there is a countable sequence of Borel uniformizations of P whose graphs cover P . More generally, this holds whenever E, F are smooth, meaning that the quotient spaces X/E, Y /F are standard Borel (c.f. Theorem 3.2.3). When E, F are not smooth, this is no longer the case. Consider for example the eventual equality relation E0 on 2N , xE0 y ⇐⇒ ∃n∀k ≥ n(xk = yk ). 111 This is a non-smooth countable Borel equivalence relation, meaning its equivalence classes are all countable. The image of E0 ⊆ 2N × 2N in 2N /E0 × 2N is Borel, has countable sections, and does not admit a Borel proper quasi-uniformization. There are a few ways one may consider generalizing the Lusin–Novikov Theorem to quotients, which are all equivalent when E, F are smooth but are not equivalent in general. Along one axis, we may consider either the existence of a single Borel uniformization of P , or a cover of P by graphs of countably many Borel uniformizations. In another direction, one may consider either Borel uniformizations or Borel proper quasi-uniformizations (though we note that these are equivalent when F is smooth). These questions were studied by de Rancourt and Miller in [dRM], and by Miller in [Mild]. We summarize their results below, in the context of uniformizations over quotients. Before we do so, we need a few definitions. Let E, F be Borel equivalence relations on Polish spaces X, Y . We let E × F denote the equivalence (x, y)(E × F )(x0 , y 0 ) ⇐⇒ xEx0 & yF y 0 on X × Y . For B ⊆ X, let [B]E = {x ∈ X : ∃x0 ∈ B(xEx0 )} denote the (E-)saturation of B, and say B is E-invariant if B = [B]E . A homomorphism from E to F is a map f : X → Y such that xEx0 =⇒ f (x)F f (x0 ), a reduction from E to F is a map f : X → Y such that xEx0 ⇐⇒ f (x)F f (x0 ), and an embedding is an injective reduction. For n ∈ N, we let Fn denote the equivalence relation on 2N given by  xFn y ⇐⇒ ∃n∀k ≥ n   X i<k xi ≡ X yi (mod n) . i<k We let I(X) = X × X be the trivial equivalence relation on any set X and ∆(X) denote the identity on X. Finally, we say a Borel equivalence relation E on a Polish space X is strongly idealistic if there is an E-invariant assignment X 3 x 7→ Ix of σ-ideals on X to points in X that is strongly Borel-on-Borel, meaning that {(z, x) : R(z,x) ∈ Ix } is Borel for all Borel sets R ⊆ 2N × X 2 . Theorem 4.1.2 ([dRM, Theorem 4.11]). Let E, F be Borel equivalence relations on Polish spaces X, Y with F strongly idealistic. Suppose that P ⊆ X/E × Y /F is Borel and has countable non-empty sections. Then exactly one of the following holds: 1. There is a countable sequence of Borel quasi-uniformizations of P that cover P . 2. There are continuous embeddings πX : 2N → X of E0 into E and πY : 2N → Y of ∆(2N ) into F such that (πX × πY )(E0 ) ⊆ P̃ . 112 Theorem 4.1.3 ([dRM, Theorem 2]). Let E, F be Borel equivalence relations on Polish spaces X, Y with F strongly idealistic. Suppose that P ⊆ X/E × Y /F is Borel and has countable non-empty sections. Then exactly one of the following holds: 1. There is a countable sequence of Borel uniformizations of P whose graphs cover P. 2. There are continuous embeddings πX : 2N → X of E0 into E and πY : 2N → Y of F into F such that (πX × πY )(E0 ) ⊆ P̃ , for some F ∈ {∆(2N )} ∪ {Fp : p prime}. This fully characterizes generalization of Lusin–Novikov for quotients when considering covers by (quasi)-uniformizations, in the case where F is strongly idealistic; see also [dRM, Theorem 3.8] for proper quasi-transversals when P has sections of bounded finite cardinality. For the existence of a single quasi-uniformization, Miller has shown the following: Theorem 4.1.4 ([Mild, Theorem 2.1]). Let E, F be Borel equivalence relations on Polish spaces X, Y with F countable. Suppose that P ⊆ X/E × Y /F is Borel and has countable non-empty sections. Then exactly one of the following holds: 1. There is a Borel quasi-uniformization of P . 2. There are continuous embeddings πX : 2N × N → X of E0 × I(N) into E and πY : 2N × N → Y of ∆(2N × N) into F such that P̃πX (z) = [πY ([z]E0 ×I(N) )]F for z ∈ 2N × N. We consider next dichotomies characterizing the canonical obstructions to the existence of a single Borel uniformization. In order to do so, we must first look at Borel cocycles on Borel equivalence relations, and their essential values. Let E be an equivalence relation on X and Γ be a group. A cocycle from E to Γ is a map ρ : E → Γ satisfying ρ(x, y)ρ(y, z) = ρ(x, z) for xEyEz. If F is an equivalence relation on Y and σ : F → Γ is a cocycle, then a homomorphism from ρ to σ is a homomorphism f from E to F such that 113 ρ(x, y) = σ(f (x), f (y)). An embedding from ρ into σ is a homomorphism from ρ to σ that is also an embedding from E into F . Given an analytic equivalence relation E on a Polish space X, a countable discrete group Γ, and a Borel cocycle ρ : E → Γ, we say that a set Λ ⊆ Γ is an essential value of ρ if Λ 6= ∅ and for every cover of X by Borel sets (Bn )n∈N , there is some n such that Λ ⊆ ρ(EBn \ ∆(Bn )). We say a sequence λ ∈ ΛN is a redundant enumeration of Λ if every element of Λ appears in this sequence infinitely often. For λ ∈ ΛN and s ∈ 2<N , let λs = Q s(i) and define ρλ : E0 → Γ by i<|s| λ(i) ρλ (s_ x, t_ x) = λs (λt )−1 for s, t ∈ 2<N , x ∈ 2N , where _ denotes concatenation of sequences. Then ρλ is a Borel cocycle with values in hΛi, and if λ is a redundant enumeration of Λ then it has Λ as an essential value by [Mila, Proposition 1.5]. Miller has characterized the essential values of Borel cocycles into countable discrete groups: Theorem 4.1.5 ([Mila, Theorem 1]). Let Λ ≤ Γ be countable discrete non-trivial groups, λ be a redundant enumeration of Λ, X be a Polish space, E be an analytic equivalence relation on X, and ρ : E → Γ be a Borel cocycle. Then the following are equivalent: 1. Λ is an essential value of ρ. 2. There is a continuous embedding of ρλ into ρ. We return now to the existence of Borel uniformizations. Given n ∈ N, let Sn denote the group of permutations of n. For Λ ≤ Sn and λ ∈ ΛN , let E0,λ denote the equivalence relation on 2N × n given by (x, i)E0,λ (y, j) ⇐⇒ xE0 y & ρλ (x, y)(j) = i. We let E0,Λ denote E0,λ for some redundant enumeration of Λ; this does not depend on the choice of λ up to isomorphism by [Mila, Theorem 5] (and in fact, it depends only on the conjugacy class of Λ). The following were proved independently by the author and Miller (personal communication), who pointed out that they follow from Theorem 4.1.5. 114 Theorem 4.1.6. Let E, F be Borel equivalence relations on Polish spaces X, Y and fix n ≥ 2. For any Borel set P ⊆ X/E × Y /F whose sections have size n, exactly one of the following holds: 1. There is a Borel uniformization of P . 2. There is a continuous reduction πX : 2N × n → X of E0 × I(n) to E and a continuous embedding πY : 2N × n → Y of E0,Λ into F such that (πX × πY )(E0 × I(n)) ⊆ P̃ , for some minimal fixed-point-free Λ ≤ Sn . Theorem 4.1.7. Let E, F be Borel equivalence relations on Polish spaces X, Y with F strongly idealistic and fix n ≥ 2. For any Borel set P ⊆ X/E × Y /F whose sections are non-empty and have size ≤ n, exactly one of the following holds: 1. There is a Borel uniformization of P . 2. There are minimal fixed-point-free Λ ≤ Sk for k ≤ n, a continuous reduction πX : 2N × k → X of E0 × I(k) to E and a continuous embedding πY : 2N × k → Y of E0,Λ into F such that P̃πX (z) = [πY ([z]E0 ×I(k) )]F for z ∈ 2N × k. Theorem 4.1.8. Let E, F be Borel equivalence relations on Polish spaces X, Y and fix n ≥ 2. For any Borel set P ⊆ X/E × Y /F whose sections have size n, exactly one of the following holds: 1. There is a Borel proper quasi-uniformization of P . 2. There is a continuous reduction πX : 2N × n → X of E0 × I(n) to E and a continuous embedding πY : 2N × n → Y of E0,Λ into F such that (πX × πY )(E0 × I(n)) ⊆ P̃ , for some minimal transitive Λ ≤ Sn . Theorem 4.1.9. Let E, F be Borel equivalence relations on Polish spaces X, Y with F strongly idealistic and fix n ≥ 2. For any Borel set P ⊆ X/E × Y /F whose sections are non-empty and have size ≤ n, exactly one of the following holds: 1. There is a Borel proper quasi-uniformization of P . 115 2. There are minimal transitive Λ ≤ Sk for k ≤ n, a continuous reduction πX : 2N × k → X of E0 × I(k) to E and a continuous embedding πY : 2N × k → Y of E0,Λ into F such that P̃πX (z) = [πY ([z]E0 ×I(k) )]F for z ∈ 2N × k. For Borel equivalence relations E, F on X, Y and P ⊆ X/E × Y /F Borel, we say that P has bounded finite sections if the cardinality of the sections of P have a finite upper bound, and P has σ-bounded finite sections if there are Borel sets Pn ⊆ P S with bounded finite sections such that P = n Pn . We note the following: Proposition 4.1.10. Let E, F be Borel equivalence relations on Polish spaces X, Y , and let P ⊆ X/E × Y /F be Borel. The following are equivalent: 1. P has σ-bounded finite sections. 2. P̃ can be covered by countably many Borel sets P̃n ⊆ P̃ , not necessarily E × F invariant, for which P̃n /(E × F ) has bounded finite sections. Moreover, if F is strongly idealistic then this is also equivalent to 3. P can be covered by countably many Borel sets Pn ⊆ P such that for all n, the non-empty sections of Pn have the same finite cardinality. Note that if P has σ-bounded finite sections and F is strongly idealistic, then by [dRM, Theorem 2.12] there is a Borel quasi-uniformization of P . In order to characterize the existence of Borel uniformizations when P has σ-bounded finite sections, we extend Miller’s characterization of essential values of Borel cocycles to countable products of finite groups. Given a sequence Γ = (Γi )i∈N of countable groups, let Γ̃k = i≤k Γi for k ≤ ∞. We abuse notation and let proji denote the projections Γ̃k → Γ̃i for i ≤ k ≤ ∞. Q Call a sequence of subgroups Λi ≤ Γ̃i coherent if proji (Λk ) = Λi for i ≤ k < ∞. Given a coherent sequence Λ, we say a sequence λi ∈ Λi is a redundant enumeration of Λ if {proji (λk ) : k ≥ i} is a redundant enumeration of Λi for all i ∈ N. Note that every coherent sequence admits a redundant enumeration. 116 Suppose that F k is a family of subsets of Γk for k ∈ N. We let F̃ k denote the family of subgroups Λ ≤ Γ̃k for which projΓi (Λ) contains an element of F i for all i ≤ k. Suppose now that E is an analytic equivalence relation on a Polish space X, Γ is a sequence of countable groups, F i is a family of subsets of Γi , and ρ : E → Γ̃∞ is a Borel cocycle. We say F is an essential value of ρ if for every cover (Bi,k )i,k∈N of X by Borel sets, there are i, k ∈ N for which projΓi (ρ(EBi,k \ ∆(Bi,k )) contains an element of F i . Finally, given a sequence of countable groups Γ, λi ∈ Γ̃i , and a Borel cocycle ρ : E0 → Γ̃∞ , we say ρ is consistent with λ if projk (ρ(0k_ 1_ x, 0k_ 0_ x)) = λk for all k ∈ N, x ∈ 2N . The following is an analogue of Theorem 4.1.5 for cocycles into countable products of finite groups. Theorem 4.1.11. Let Γ be a sequence of finite groups and F i be a family of subsets of Γi that is closed under conjugation, for which every set in F i contains a non-identity element. Let E be an analytic equivalence relation on a Polish space X and ρ : E → Γ̃∞ be a Borel cocycle. Then the following are equivalent: 1. The family F is an essential value for ρ. 2. There is a coherent sequence Λi ∈ F̃ i , a redundant enumeration λ of Λ, a Borel cocycle ρ : E0 → Γ̃∞ consistent with λ, and a continuous embedding of ρ into ρ. Given a sequence α(n) ≥ 2, n ∈ N, let Γαi = Sα(i) . Let also n(α, i) = Q Γ̃αi = l≤i Sα(l) ≤ Sn(α,i) for i ≤ ∞, and write n(α, −1) = 0. l≤i α(l), so that P For a cocycle ρ : E0 → S∞ , let E0,ρ denote the equivalence relation on 2N given by (x, i)E0,ρ (y, j) ⇐⇒ xE0 y & ρ(x, y)(j) = i. Using Theorem 4.1.11, we prove the following: Theorem 4.1.12. Let E, F be Borel equivalence relations on Polish spaces X, Y . Let Pn ⊆ X/E × Y /F be pairwise-disjoint Borel sets with sections of cardinality exactly S α(n) ≥ 2, and P = n Pn . Then exactly one of the following holds: 117 1. There is a Borel uniformization of P . 2. There is a coherent sequence of fixed-point-free subgroups Λi ≤ Γ̃αi , a redundant enumeration λ of Λ, a Borel cocycle ρ : E0 → Γ̃α∞ consistent with λ, and continuous reductions πX : 2N × N → X of E0 × I(N) to E and πY : 2N × N → Y of E0,ρ to F , such that: a) P̃πX (z) = [πY ([z]E0 ×I(N) )]F for z ∈ 2N × N; and b) πY is an embedding when restricted to S i∈N N0i × (n(α, i) \ n(α, i − 1)). We also prove a parametrized version of Theorem 4.1.11, and use this to give the following characterization of the existence of Borel (quasi-)uniformizations in the case of σ-bounded finite index. Theorem 4.1.13. Let E, F be Borel equivalence relations on Polish spaces X, Y with F strongly idealistic. For any Borel set P ⊆ X/E × Y /F with non-empty σ-bounded finite sections, exactly one of the following holds: 1. There is a Borel uniformization of P . 2. One of the following holds: a) There are minimal fixed-point-free Λ ≤ Sk for k ∈ N, and continuous embeddings πX : 2N × k → X of E0 × I(k) into E and πY : 2N × k → Y of E0,Λ into F such that P̃πX (z) = [πY ([z]E0 ×I(k) )]F for z ∈ 2N × k. b) There is some sequence α(n) ≥ 2, a coherent sequence of fixed-pointfree subgroups Λi ≤ Γ̃αi , a redundant enumeration λ of Λ, a Borel cocycle ρ : E0 → Γ̃α∞ consistent with λ, and continuous reductions πX : 2N ×N → X of E0 × I(N) to E and πY : 2N × N → Y of E0,ρ to F , such that: i. P̃πX (z) = [πY ([z]E0 ×I(N) )]F for z ∈ 2N × N; and ii. πY is an embedding when restricted to S i∈N N0i × (n(α, i) \ n(α, i − 1)). It is unclear whether we can ensure that πX , πY are embeddings in Theorem 4.1.12 (2) or Theorem 4.1.13 (2b). 118 Uniformization One σ-bounded finite sections Cover All [dRM] Proper quasi-uniformization 1. All, for F countable [Mild] 2. σ-bounded finite sections All [dRM] Table 4.1: A summary of Lusin–Novikov dichotomies over quotients. We summarize what is known about Lusin–Novikov Theorems over quotients in Table 4.1. In this table, we assume F is strongly idealistic, and we show for which sets P there is a dichotomy theorem. We prove Theorem 4.1.11 in Section 4.2, as well as a parametrized version. We then give proofs of Theorems 4.1.6 to 4.1.9, 4.1.12, and 4.1.13 and Proposition 4.1.10 in Section 4.3. Remark 4.1.14. We have chosen to state and prove these results only for Borel equivalence relations on Polish spaces. However, we note that our proofs actually show that Theorems 4.1.6, 4.1.8, and 4.1.12 hold more generally in the case where X, Y are Hausdorff spaces, E is analytic, and P̃ ⊆ X × Y is analytic and E × ∆(Y )-invariant, assuming that alternative 1 is adequately modified; see e.g. [dRM, Theorem 3.8, Theorem 4.11] and [Mild, Theorem 1]. Acknowledgements. Research partially supported by NSF Grant DMS-1950475. We would like to thank Jan Grebík for asking the question that motivated this work, and Ben Miller for their feedback, helpful comments, and for pointing out the connection between uniformization problems and Borel cocycles. Thanks also to Alexander Kechris and Esther Nam for the encouragement and support. 4.2 4.2.1 Dichotomies for essential values into pro-finite groups Technical preliminaries Let Γ be a sequence of countable groups, Λi ≤ Γ̃i be a coherent sequence of subgroups, and λ be a redundant enumeration of Λ. For γ ∈ Γ̃∞ , we let γΛγ −1 , γλγ −1 denote the sequences (proji (γ)Λi proji (γ)−1 )i , (proji (γ)λi proji (γ)−1 )i . Note that γΛγ −1 (resp.γλγ −1 ) are coherent (resp. a redundant enumeration of γΛγ −1 ). Proposition 4.2.1. Let Γ be a sequence of countable groups, Λi ≤ Γ̃i be a coherent sequence of subgroups, λ be a redundant enumeration of Λ, and ρ : E0 → Γ̃∞ be a Borel cocycle consistent with λ. 119 Suppose that Dn ⊆ 2N × 2N are closed and nowhere dense and R ⊆ 2N × 2N is meagre. Then there is some γ ∈ Γ̃∞ , a Borel cocycle ρ̃ : E0 → Γ̃∞ consistent with γλγ −1 , and a continuous embedding π : 2N → 2N of ρ̃ into ρ such that for x, y ∈ 2N , x(n) 6= y(n) =⇒ π(x) D E0 y =⇒ π(x)Rπ(y). n π(y) & x Proof. Let Un be a decreasing sequence of symmetric dense open sets such that T Dn ∩ Un = R ∩ n Un = ∅. For all s ∈ 2≤N and n ≤ |s|, let zn (s) denote the sequence where we replace the first n elements of s by 0, i.e., zn (s)(k) = s(k) for n ≤ k < |s| and zn (s)(k) = 0 for k < n. For n, k ∈ N and s ∈ 2k , let λ0 s = i<k projn (λn+i )s(i) ∈ Λn ⊆ Γ̃n . Note that there is some ambiguity as to which n is chosen in this definition, as s may begin with 0, though all such choices will be consistent with taking projections. In what follows, which n is chosen will be clear from context. n_ Q For all n, k ∈ N, x ∈ 2N and u, v ∈ 2k we have n_ u projn (ρ(0n_ u_ x, 0n_ v _ x)) = λ0 n_ v (λ0 )−1 . (†) To see this, note that it suffices to show this for v = 0k . Arguing inductively, suppose we have shown this for all u ∈ 2k and note that projn (ρ(0n_ u_ i_ x, 0n_ 0k_ 0_ x)) = projn (ρ(0n_ u_ i_ x, 0n_ 0k_ i_ x)) · projn (ρ(0n_ 0k_ i_ x, 0n_ 0k_ 0_ x)) =λ0 n_ u =λ0 n_ u_ i (projn (λn+k ))i for all u ∈ 2k , i ∈ 2, where the second equality follows from our inductive hypothesis and the fact that ρ is consistent with λ. We will recursively construct un , ti,n ∈ 2<N , An , Vn,l ⊆ 2N , γn ∈ Γ̃n , δn ∈ Λn , and φn : 2n → 2<N for i ∈ 2, n, l ∈ N so that for all n ∈ N, n (a) |t0,n | = |t1,n | > 0, t0,n 6= t1,n , φn+1 (s_ i) = φn (s)_ u_ n ti,n for all i ∈ 2, s ∈ 2 , and φ0 (∅) = ∅ (so in particular |φn (s)| ≥ n for n ∈ N); (b) Nφn (s0 )_ u_ × Nφn (s1 )_ u_ ⊆ Un for all (s0 , s1 ) ∈ 2n × 2n ; n t0,n n t1,n (c) An is comeagre in Nφn (0n )_ un and (Vn,l )l∈N is a decreasing sequence of dense T open sets in Nφn (0n )_ un satisfying l Vn,l ⊆ An ; 120 (d) Nφn (0m_ s)_ u_ ⊆ Vm,n−m for all m ≤ n, s ∈ 2n−m and i ∈ 2; n ti,n (e) projn (ρ(x, zn (x))) = γn for all x ∈ An ; (f) λzn (φn (0 )) un t1,n (λzn (φn (0 )) un t0,n )−1 = δn λn δn−1 ; and n _ _ n _ _ (g) projn−1 (γn δn ) = γn−1 δn−1 for n > 0. To this end, suppose we have completed the construction below n and define φn as in (a). Let γx = projn (ρ(x, zn (x))) for x ∈ 2N and let Aγ = {x ∈ 2N : γ = γx } for γ ∈ Γ̃n . The sets Aγ are Borel and cover 2N , so there is some γn such that An = Aγn is non-meagre in Nφn (0n ) . Let un ∈ 2<N be such that An is comeagre in Nφn (0n )_ un , and fix a decreasing sequence of dense open sets Vn,l in Nφn (0n )_ un satisfying (c). Now recursively construct t0i,n ∈ 2<N such that (b) holds with t0i,n replacing ti,n . This is possible as Un is dense and open. We then recursively extend the t0i,n to some t00i,n such that (d) holds with t00i,n replacing ti,n . This is possible as the Vm,n−m are dense and open in Nφm (0m )_ um and φn (0m_ s)_ un ⊇ φm (0m )_ um . We can make the t00i,n longer to ensure that |t000,n | = |t001,n |. If n = 0, we let δ0 = 1Λ0 . Otherwise, we note that An−1 is comeagre in Nφn (0n ) , so there is some x ∈ An ∩ An−1 , which implies that projn−1 (γn ) = projn−1 (ρ(x, zn (x))) = projn−1 (ρ(x, zn−1 (x))ρ(zn−1 (x), zn (x))) x(n−1) = γn−1 λn−1 by (†), so projn−1 (γn ) = γn−1 δn−1 δ for some δ ∈ Λn−1 . Choose δn ∈ Λn satisfying projn−1 (δn ) = δ −1 , which is possible as Λ is a coherent sequence. This ensures that (g) is satisfied. Finally, let λ = (λzn (φn (0 )) un t1,n )−1 δn λn δn−1 λzn (φn (0 )) un t0,n ∈ Λn ⊆ Γ̃n . Find m such that projn (λ|φn (0n )|+|un |+|t00i,n |+m ) = λ, and let ti,n = t00i,n _ 0m_ (i). Then n n _ _ _ _ 00 n _ _ n n _ _ 00 _ _ 00 n _ _ 00 λzn (φn (0 )) un t1,n (λzn (φn (0 )) un t0,n )−1 = λzn (φn (0 )) un t1,n λ(λzn (φn (0 )) un t0,n )−1 = δn λn δn−1 , so (f) holds. This completes the recursive construction. Now define π(x) = n φn (xn) for x ∈ 2N . By (a), π : 2N → 2N is a continuous embedding of E0 into itself. By (b), if x E0 y then π(x)Rπ(y), and if x(n) 6= y(n) then S 121 π(x) Di π(y) for i ≤ n. We claim that (‡) projn (ρ(π(0n_ 1_ x), π(0n_ 0_ x))) = γn δn λn δn−1 γn−1 for all n ∈ N, x ∈ 2N . To see this, note that by (d) we have π(0n_ i_ x) ∈ Vn,l for all l, hence by (c) π(0n_ i_ x) ∈ An . By (e), projn (ρ(π(0n_ i_ x), zn (π(0n_ i_ x)))) = γn . Thus it suffices to check that projn (ρ(zn (π(0n_ 1_ x)), zn (π(0n_ 0_ x)))) = δn λn δn−1 . _ To see this, note that there is some y ∈ 2N such that π(0n_ i_ x) = φn (0n )_ u_ n ti,n y for i ∈ 2, so n _ _ n _ _ projn (ρ(zn (π(0n_ 1_ x)), zn (π(0n_ 0_ x)))) = λzn (φn (0 )) un t1,n (λzn (φn (0 )) un t0,n )−1 = δn λn δn−1 by (f) and (†). By (g) the sequence (γn δn )n∈N is coherent, and therefore corresponds to some γ ∈ Γ̃∞ . If ρ̃ is the pullback of ρ along π, then ρ̃ is consistent with γλγ −1 by (‡), so π is the desired embedding. Proposition 4.2.2. Let Γ be a sequence of countable groups, Λi ≤ Γ̃i be a coherent sequence of subgroups, λ be a redundant enumeration of Λ, and ρ : E0 → Γ̃∞ be a Borel cocycle consistent with λ. Suppose B ⊆ 2N is Baire measurable and non-meagre. Then proji (ρ(E0 B \ ∆(B))) contains a conjugate of Λi for all i ∈ N. Proof. Fix i ∈ N. For all x ∈ 2N , let γx = proji (ρ(x, zi (x))), where zi : 2N → 2N is the function which replaces the first i elements of x with 0. Let Aγ = {x ∈ 2N : γ = γx } for γ ∈ Γ̃i , and note that these sets are Baire-measurable and cover 2N , so there is some γ for which Aγ ∩ B is non-meagre. Let s ∈ 2<N be such that Aγ ∩ B is Q comeagre in Ns and |s| ≥ i, and let λ = i≤n<|s| proji (λn )s(n) ∈ Λi . As in the proof of Proposition 4.2.1, we see that for n ∈ N and x ∈ 2N satisfying s_ 0n_ j _ x ∈ Aγ for j ∈ 2 we have proji (ρ(s_ 0n_ 1_ x, s_ 0n_ 0_ x)) = γλ proji (λ|s|+n )λ−1 γ −1 . Since λ is a redundant enumeration of Λ and λ ∈ Λi , {γλ proji (λ|s|+n )λ−1 γ −1 } is a redundant enumeration of γΛi γ −1 , and since Aγ ∩ B is comeagre in Ns we can find for all n ∈ N some x ∈ 2N so that s_ 0n_ j _ x ∈ Aγ ∩ B for j ∈ 2. It follows that γΛi γ −1 ⊆ proji (ρ(E0 B \ ∆(B))). 122 4.2.2 The G0 dichotomy for sequences of graphs We recall now Miller’s G0 dichotomy for sequences of graphs. We say S ⊆ 2<N is sparse if it contains at most one sequence of every length, and dense if for all t ∈ 2<N there is some s ∈ S with t ⊆ s. We say a sequence S of subsets of 2<N is sparse if its union is sparse, and dense if every element of the sequence is dense. For s ∈ 2<N , let Gs be the directed graph Gs = {(s_ 0_ x, s_ 1_ x) : s ∈ S & x ∈ 2N }. For S ⊆ 2<N , we write GS = GS = (GSi )i∈N . S <N we let s∈S Gs , and for a sequence S of subsets of 2 If G, H are sequences of directed graphs on X, Y , a homomorphism from G to H is a map f : X → Y such that for all i ∈ N, xGi x0 =⇒ f (x)Hi f (x0 ). We say a set B ⊆ X is G-independent if it is Gi -independent for some i ∈ N. If X is a Polish space, we write χB (G) ≤ ℵ0 if there is a cover of X by countably many Borel G-independent Borel sets. Theorem 4.2.3 (The G0 dichotomy for sequences, [Mil12, Theorem 21]). Let G be a sequence of analytic directed graphs on a Polish space X and S be a sparse dense sequence of subsets of 2<N . Then exactly one of the following holds: 1. χB (G) ≤ ℵ0 . 2. There is a continuous homomorphism from GS to G. 4.2.3 Proof of Theorem 4.1.11 To see that 2 implies 1, fix Λ, λ, ρ, π satisfying 2 and let Bi,k be a cover of X by Borel sets. Then there are i, k so that B = π −1 (Bi,k ) is non-meagre, so by Proposition 4.2.2 there is some γ ∈ Γ̃i with γΛi γ −1 ⊆ proji (ρ(E0 B\∆(B))) = proji (ρ(EBi,k \∆(Bi,k ))). It follows that this set contains an element of F i , as F i is closed under conjugation. We now show that 1 implies 2. Let T be the tree of all sequences of the form (proj0 (Λ), . . . , proji (Λ)) for i ≤ n < N and Λ ∈ F̃ n . Since the groups Γ̃i are finite, T is finitely branching. For n ∈ N let Tn = {t(n) : t ∈ T & |t| > n} be the n-th level of T , and note that each S Tn is finite. Let (Ank )k<Kn be an enumeration of all subsets of Tn which intersect every element of Tn and let Gn,k = (projn ◦ρ)−1 (Ank ) \ ∆(X). 123 Suppose B ⊆ X is Gn,k -independent. Then projn (ρ(EB \ ∆(B))) does not contain any element of Tn . By [Mila, Proposition 1.4], we may partition B into countably many Borel sets Bl so that projn (ρ(EBl \ ∆(Bl ))) does not generate a group containing an element of Tn . In particular, for all l there is some i so that projΓi (ρ(EBl \ ∆(Bl ))) does not contain an element of F i . It follows that if χB (G) ≤ ℵ0 , then there is a cover Bi,k of X so that projΓi (ρ(EBi,k \ ∆(Bi,k ))) does not contain an element of F i , and in particular F is not an essential value of ρ. Fix now a sparse dense sequence (Sn,k )n,k∈N of subsets of 2<N . By Theorem 4.2.3, we may assume that there is a continuous homomorphism φ : 2N → X from GS to G. Claim 4.2.4. There is a coherent sequence Λ which is a path through T , a redundant enumeration λ of Λ, a Borel cocycle ρ : E0 → Γ̃∞ consistent with λ, and a continuous homomorphism ψ : 2N → X of ρ to ρ. We postpone the proof of the claim to the end. Suppose now that we have Λ, λ, ρ, ψ as in the claim, and let D = (ψ × ψ)−1 (∆(X)), E 0 = (ψ × ψ)−1 (E), and xF 0 x0 ⇐⇒ proj0 (ρ(ψ(x), ψ(x0 ))) = 1Γ0 . We claim that E 0 is meagre. Given this, D ⊆ E 0 is closed and nowhere dense, so by Proposition 4.2.1 there is some γ ∈ Γ̃∞ , a cocycle ρ̃ consistent with γλγ −1 and a continuous embedding π : ρ̃ → ρ which is moreover a homomorphism from (∼∆(2N ), ∼E0 ) to (∼D, ∼E 0 ). It follows that ψ ◦ π witnesses alternative 2 of Theorem 4.1.11. To see that E 0 is meagre, we first show that every F 0 -class is meagre. Indeed, let C be an F 0 -class and note that if x, y ∈ C and xE0 y then proj0 (ρ(x, y)) = 1. By Proposition 4.2.2 and our assumption that Λ0 ∈ F̃ 0 is non-trivial, C is meagre. Next, note that every E 0 -class is a finite union of F 0 -classes. To see this, let C be an E 0 -class and let x0 , . . . , xn−1 ∈ C be a sequence of maximal length such that proj0 (ρ(ψ(x0 ), ψ(xi ))) are distinct for all i < n (such a sequence has length at most |Γ0 |). If y ∈ C, let i < n be such that proj0 (ρ(ψ(x0 ), ψ(y))) = proj0 (ρ(ψ(x0 ), ψ(xi ))). Then proj0 (ρ(ψ(y), ψ(xi ))) = 1, so yF 0 xi , and hence C = [x0 ]F 0 ∪ · · · ∪ [xn−1 ]F 0 . Thus every E 0 -class is meagre, and hence so is E 0 by the Kuratowski-Ulam Theorem [Kec95, 8.41]. Finally, it remains to prove the claim. Proof of claim. For n ∈ N, λ ∈ Γ̃n , s ∈ 2<N , let Cλ,s = {x ∈ 2N : projn (ρ(φ(s_ 1_ x), φ(s_ 0_ x))) = λ} 124 and set Sλ = {s ∈ 2<N : Cλ,s is non-meagre}. For n ∈ N, Λ ∈ Tn , u ∈ 2<N , write A(Λ, u) ⇐⇒ ∀v ∈ 2<N ∃∞ m ≥ n∃Λ0 ∈ Tm (projn (Λ0 ) = Λ & ∀λ ∈ Λ0 ∃w ∈ 2<N (Sλ is dense below u_ v _ w)). Here, ∃∞ means “there exists infinitely many”. For all n ∈ N, k < Kn , s ∈ Sn,k we S have 2N = λ∈Ank Cλ−1 ,s , since φ is a homomorphism from GSn,k to Gn,k . Note also that if λ ∈ Γ̃m and n ≤ m, then Sλ ⊆ Sprojn (λ) . In particular, if A(Λ, u) then Sλ is dense below u for all λ ∈ Λ. Indeed, for v ∈ 2<N we may fix Λ0 ∈ Tm , m ≥ n so that projn (Λ0 ) = Λ and ∀λ0 ∈ Λ0 there is some element of Sλ0 below u_ v, so this holds also for Sλ for all λ ∈ Λ. Finally, note that A(Λ, u) =⇒ A(Λ, u_ v) for all Λ and u, v ∈ 2<N . Subclaim 4.2.5. Let n ∈ N, Λ ∈ Tn , u ∈ 2<N . Then A(Λ, u) =⇒ ∀v ∈ 2<N ∃w ∈ 2<N ∃Λ0 ∈ Tn+1 (projn (Λ0 ) = Λ & A(Λ0 , u_ v _ w)). Also, A(Λ, u) holds for some u ∈ 2<N , Λ ∈ T0 . Proof. We show the contrapositive. Suppose there is some u0 extending u such that ∀Λ0 ∈ Tn+1 ∀v ∈ 2<N (projn (Λ0 ) = Λ =⇒ ¬A(Λ0 , u0 _ v)), in order to show that ¬A(Λ, u). Let Λ0 , . . . , Λl−1 be an enumeration of Tn+1 , and recursively construct u0 ⊆ u1 ⊆ · · · ⊆ ul and N1 , . . . , Nl ∈ N as follows: Given ui , i < l, if projn (Λi ) 6= Λ0 we let Ni+1 = n and ui+1 = ui . Otherwise, we have ¬A(Λi , ui ) by our assumption, so we may fix ui+1 extending ui and Ni+1 > n so that for all m ≥ Ni+1 and Λ0 ∈ Tm , projn+1 (Λ0 ) = Λi =⇒ ∃λ ∈ Λ0 ∀w ∈ 2<N (Sλ is not dense below ui+1 _ w). Let N = maxi≤l Ni . For all m ≥ N, Λ0 ∈ Tm , if projn (Λ0 ) = Λ then projn+1 (Λ0 ) = Λi for some i, so by our construction we have ∃λ ∈ Λ0 ∀w ∈ 2<N (Sλ is not dense below ui+1 _ w) and in particular this holds for ul replacing ui+1 . Thus ul , N witness that ¬A(Λ, u). 125 Suppose now for the sake of contradiction that ¬A(Λ, u) for all u ∈ 2<N , Λ ∈ T0 . As above, by enumerating T0 we can recursively construct some u ∈ 2<N and some N so that ∀Λ ∈ TN ∃λ ∈ Λ∀v ∈ 2<N (Sλ is not dense below u_ v). Let A ⊆ TN be the set of all λ ∈ TN such that Sλ−1 is not dense below u_ v for any v ∈ 2<N . Since A is finite, we can recursively build some v ∈ 2<N so that Sλ−1 contains no elements extending u_ v for all λ ∈ A. By construction A intersects every element of TN , so A = AN k for some k < KN , and by the density of SN,k there is some S s ∈ SN,k extending u_ v. Now 2N = λ∈A Cλ−1 ,s , so Cλ−1 ,s is non-meagre for some λ ∈ A. But then s ∈ Sλ−1 , a contradiction. // S S Fix a bijection p = (p0 , p1 ) : N → N2 satisfying p0 (n) ≤ n for all n. We will recursively construct un ∈ 2<N , Λn ∈ Tn , λn , λn,l ∈ Λn , Vn,l ⊆ 2<N , ψn : 2n → 2<N for n, l ∈ N satisfying: (a) ψ0 (∅) = u0 and ψn+1 (s_ i) = ψn (s)_ i_ un+1 for s ∈ 2n , i ∈ 2; (b) ψn (0n ) ∈ Sλn ; (c) Cλn ,ψn (0n ) is comeagre in Nun+1 , Vn,l is a decreasing sequence of sets which are T dense and open in Nun+1 and satisfy l Vn,l ⊆ Cλn ,ψn (0n ) ; (d) Nun+1 ⊆ Vn,0 , and Nt_ i_ un+1 ⊆ Vm,n−m for all m < n, s ∈ 2n−m , i ∈ 2, where t ∈ 2<N is such that ψn (0m_ s) = ψm (0m )_ j _ t for some j ∈ 2; (e) A(Λn , ψn (0n )); (f) projn (Λn+1 ) = Λn and (λn,l )l∈ω is a redundant enumeration of Λn ; (g) projp0 (n) (λn ) = λp(n) . We will construct these sequences in stages, so that at stage n of the construction we will have defined uk , Λk , λk , λk,l , ψk for k ≤ n, and Vk,l for k < n. We begin with stage n = 0. By the sublemma, there are Λ0 ∈ T0 and u ∈ 2<N satisfying A(Λ0 , u). Let (λ0,l )l be a redundant enumeration of Λ0 and let λ0 = λp(0) . By the remarks preceding the sublemma, Sλ0 is dense below u, so we may choose u0 ∈ Sλ0 extending u and set ψ0 (∅) = u0 . Note that A(Λ0 , u0 ) by the remarks preceding the sublemma. 126 Now suppose we have completed stage n of the construction. By (b), there is some u such that Cλn ,ψn (0n ) is comeagre in Nu . Fix Vn,l as in (c) (with u taking the place of un+1 ). By (e) and the sublemma, we can (after possibly extending u) find some Λn+1 ∈ Tn+1 so that projn (Λn+1 ) = Λn and A(Λn+1 , ψn (0n )_ 0_ u). Let (λn+1,l )l be a redundant enumeration of Λn+1 , and let λn+1 ∈ Λn+1 satisfy (g) (which is possible as projp0 (n+1) (Λn+1 ) = Λp0 (n+1) ). By extending u, we may assume that Nu ⊆ Vn,0 . We may then recursively extend u so that the rest of (d) holds, as there are only finitely many such m < n, s ∈ 2n−m , i ∈ 2 to consider, and given these and t as in (d) we have that um+1 ⊆ t and hence by (c) that Vm,n−m is dense and open in Nt . Finally, we may extend u to some un+1 satisfying ψn (0n )_ 0_ un+1 ∈ Sλn+1 , as this set is dense below ψn (0n )_ 0_ u by the remarks preceding the sublemma. We define ψn+1 (s_ i) = ψn (s)_ i_ un+1 , and note that this completes stage n + 1 of the construction as (c), (d), (e) continue to hold when we extend u. This completes the recursive construction. Let ψ(x) = n ψn (xn). By (a), (b) this gives a continuous map 2N → 2N such that for all n ∈ N, x ∈ 2N there is some y ∈ 2N with ψ(0n_ i_ x) = ψn (0n )_ i_ un+1 _ y for T i ∈ 2. By (c), (d) we have un+1 _ y ∈ l Vn,l ⊆ Cλn ,ψn (0n ) , so S projn (ρ((φ ◦ ψ)(0n_ 1_ x), (φ ◦ ψ)(0n_ 0_ x))) = λn . It follows that if ρ is the restriction to E0 of the pullback of ρ along φ ◦ ψ, then ρ is consistent with λ and φ ◦ ψ is a continuous homomorphism of ρ to ρ. By (f), (g), Λ is a coherent path through T and λ is a redundant enumeration of Λ. / Remark 4.2.6. In the proof of the claim, we may replace A(Λ, u) with A0 (Λ, u) ⇐⇒ ∀λ ∈ Λ(Sλ is dense below u), and then prove that for all u ∈ 2<N and n ∈ N there are v ∈ 2<N and Λ ∈ Tn such that A0 (Λ, u_ v) (this is essentially the second part of the subclaim). This is enough to ensure that we can recursively construct ψ in the claim, except without the guarantee that the resulting sequence Λ is coherent. We can then build a tree from sequences of projections of elements of Λ (just as we did in the definition of T ), and use König’s lemma to find a branch through this tree, which would give a coherent sequence. One could then pre-compose ψ with an appropriate function π to complete the proof of the claim (the construction of this π being very explicit and straightforward). The definition of A and the first part of the sublemma 127 essentially “unravel” the proof of König’s lemma in this remark (this is why the “∃∞ ” quantifier appears in the definition of A). 4.2.4 A parametrized version of Theorem 4.1.11 The proof of Theorem 4.1.11 is effective, meaning that if E is a ∆11 equivalence relation on NN , Γ is a ∆11 sequence of finite groups (coded in some appropriate space), ρ : E → Γ̃∞ is a ∆11 cocycle, the family F is ∆11 , and F is not an essential value of ρ, then there is a uniformly ∆11 sequence of sets (Bi,k )i,k covering NN so that for all i, k ∈ N, projΓi (ρ(EBi,k \ ∆(Bi,k ))) does not contain an element of F i . To see this, note that Theorem 4.2.3 is effective, i.e., if G is a uniformly Σ11 sequence of graphs on NN and χB (G) ≤ ℵ0 , then G admits a uniformly ∆11 sequence of G-independent sets covering NN . Now consider the proof of Theorem 4.1.11. It is easy to see that T, Ank are (uniformly) ∆11 , and hence so is G. Thus, if F is not an essential value of ρ, there is a uniformly ∆11 sequence (Bl )l of sets covering NN that are G-independent. Note that being Gi,k independent is a Π11 -onΣ11 property, so that by the Number Uniformization Theorem [Mos09, 4B.5] there is a ∆11 map l 7→ (il , kl ) so that Bl is Gil ,kl -independent. The proof of [Mila, Proposition 1.4] is effective enough that we may partition the sets Bl to get a uniformly ∆11 sequence (Bl,j )l,j so that projil (ρ(EBl,j \ ∆(Bl,j ))) does not generate an element of Til . Then for all l, j there is some m ≤ il so that projΓm (ρ(EBl,j \ ∆(Bl,j ))) does not contain an element of F m . This is again a Π11 -on-Σ11 property, so by the Number Uniformization Theorem there is a ∆11 map (l, j) 7→ ml,j witnessing this. Using these maps, we can relabel the sets Bl,j so that we have a uniformly ∆11 sequence Bi,k witnessing that F is not an essential value of ρ. Thus, using [Mos09, 4D.4] we get the following parametrized version of Theorem 4.1.11. Theorem 4.2.7. Let X, Z be Polish spaces and D ⊆ X × Z, E ⊆ X 2 × Z be Borel such that E z is an equivalence relation on Dz for all z ∈ Z. Let FinGrp be the countable discrete space of finite groups, E(FinGrp) be the space of pairs (Γ, γ) where γ ∈ Γ ∈ FinGrp, and F (FinGrp) be the space of pairs (Γ, F) where Γ ∈ FinGrp and F is a family of subsets of Γ. Let z 7→ Γz be a Borel map from z to FinGrpN , and define Γ̃z as in the usual setting. Let ρ : E → E(FinGrp)N be a Borel map such that ρz is a cocycle from E z to Γ̃z∞ for all z ∈ Z. 128 Let z 7→ F z ∈ F (FinGrp) be a Borel map taking z ∈ Z to a family of sets satisfying the hypotheses of Theorem 4.1.11 for the equivalence relation E z , the family of finite groups Γz , and the cocycle ρz . The following are equivalent: 1. The family F is an essential value of ρ, meaning that for every cover of D by Borel sets Bi,k , there are i, k ∈ N and z ∈ Z for which z z projΓzi (ρz (E z Bi,k \ ∆(Bi,k ))) contains an element of F zi . 2. There are z ∈ Z, a coherent sequence Λi ∈ F̃ i , a redundant enumeration λ of Λ, a Borel cocycle ρ : E0 → Γ̃z∞ consistent with λ, and a continuous embedding of ρ into ρz . z Proof. By the usual transfer theorems, we may assume that X = Z = NN . By relativizing, we may also assume that D, E, Γ, ρ, F are (uniformly) ∆11 . To see that 2 implies 1, we argue as in the proof of Theorem 4.1.11. To see that 1 implies 2, suppose that 2 fails. Then F z is not an essential value of ρz for all z ∈ Z, so by the effectiveness of Theorem 4.1.11 discussed above, for all z ∈ Z there is a ∆11 (z) witness that this is the case. By [Mos09, 4D.4], there are Borel sets Bi,k ⊆ D z so that for all z ∈ Z, (Bi,k )i,k witnesses that ρz is not an essential value of F z . But then the sets Bi,k witness that 1 fails. We include below a sketch of a proof that does not use effective descriptive set theory, for the convenience of the reader. We first show that the proof of [Mila, Proposition 1.4] is uniform. Proposition 4.2.8 ([Mila, Proposition 1.4]). Let X, Z, D, E be as in Theorem 4.2.7. Let Γ : Z → FinGrp, ρ : E → E(FinGrp) be Borel maps so that ρz : E z → Γz is a cocycle for all z ∈ Z. Suppose that Λ : Z → FinGrp is a Borel map taking z ∈ Z to a subgroup of Γz such that Λz 6⊂ ρz (E z \ ∆(Dz )). Then there is a cover of D by countably many Borel sets (Bl )l∈N such that the group generated by ρz (E z Blz \ ∆(Blz )) does not contain Λz for all z ∈ Z, l ∈ N. 129 Proof. Let (Azk )k∈N be a sequence of subsets of Λz which generate Λz , and such that every generating subset appears in this sequence at least once. Let K z be such that every generating subset appears in {Azk : k < K z }. This can be done in a uniformly Borel way. We will show that for any k, there is a partition of D into countably many Borel sets (Bl )l∈N so that ρz (E z Blz \ ∆(Blz )) does not contain Azk . Suppose this has been done. Then we can let (Clz )l∈N be the partition of B z generated by each of these partitions for k < K z , and again this can easily be done in a uniformly Borel way, completing the proof. So let z 7→ Az be a Borel map so that Az generates Λz for all z. Let λz be an element of Λz which is not contained in ρz (E z B z \ ∆(B z )), and note that this can be done in a uniformly Borel by the Number Uniformization Theorem (see [Kec95, 28.5] for a classical proof). We may split Z into two Borel subsets Z0 , Z1 , in which λz is (resp. is not) the identity, and consider these cases separately. Consider first the case where z ∈ Z1 , i.e., λz is not the identity. Let wz ∈ (Az )<N be a Q word in Az satisfying i<|wz | wz (i) = λz . This can again be done in a uniformly Borel way by the Number Uniformization Theorem. By splitting Z1 according to the length of wz , we may assume that we have Borel sets Zn , n ≥ 1 so that for z ∈ Zn , |wz | = n. We may therefore consider each Zn separately. Consider now z ∈ Zn , n ≥ 1. Define recursively C0z = B z and z Ci+1 = {x ∈ Ciz : ∃y ∈ Ciz (ρz (x, y) = wz (n − 1 − i))} for all i < n. The sets Ci are clearly analytic. For z ∈ Zn , Cnz = ∅. Indeed, if xn ∈ Cnz then by reverse recursion one could find xi ∈ Ciz such that ρz (xi+1 , xi ) = wz (n − 1 − i) for i < n. But then ρz (xn , x0 ) = wz (0) . . . wz (n − 1) = λz , and since this is not the identity we must have xn 6= x0 , contradicting our choice of λz . z Thus, for z ∈ Zn , n ≥ 1 we have that ∀x, y ∈ Cn−1 (ρz (x, y) 6= wz (0)). By the First Reflection Theorem [Kec95, 35.10], we may enlarge Cn−1 into a Borel set Bn−1 for z z which this property still holds. But then ∀x, y ∈ Cn−2 \ Bn−1 (ρz (x, y) 6= wz (1)), so we may enlarge Cn−2 into a Borel set Bn−2 for which this property still holds. Continuing recursively in this way, we obtain Borel sets B0 , B1 , . . . , Bn−1 so that for all i < n and 130 x, y ∈ Biz , ρz (x, y) 6= wz (n − 1 − i). By construction, C0 ⊆ B0 ∪ · · · ∪ Bn−1 , so the sets B0 , . . . , Bn−1 form the desired partition of D restricted to Zn . Consider now z ∈ Z0 , so that the identity is not contained in ρz (E z Dz \ ∆(Dz )). If Az = {1}, then there is nothing to do, so we may assume wlog that Az 6= {1} for all z ∈ Z0 . Let γ z be an arbitrary Borel assignment of a non-identity element of Az to z ∈ Z0 . We will show that there is a partition of D ∩ X × Z0 into Borel sets (Bl )l∈N so that ρz (E z Blz \ ∆(Blz )) does not contain γ z for all z ∈ Z0 . As D ∩ X × Z0 is Borel, there is a sequence (Wl )l∈N of Borel sets in D ∩ X × Z0 which separates points. Let Clz = {x ∈ Wlz : ∃y ∈ ∼Wlz (γ z = ρz (x, y))}. Then the sets Cl are analytic, and we claim that γ z ∈ / ρz (E z Clz \ ∆(Clz )) for all z ∈ Z0 , l ∈ N. Indeed, if there were x, y ∈ Clz with ρz (x, y) = γ z , then we may fix w ∈ ∼Wlz satisfying γ z = ρz (x, w). But then ρz (y, w) = 1, so by our assumption that ρz (E z Dz \ ∆(Dz )) does not contain the identity we must have y = w, contradicting the fact that y ∈ Clz ⊆ Wlz and w ∈ / Wlz . We may therefore expand each Cl into Borel sets Bl for which ρz (E z Blz \ ∆(Blz )) does not contain γ z for all z ∈ Z0 , so it suffices to show that for all z ∈ Z0 and S S x, y ∈ ∼ l∈N Blz , ρz (x, y) 6= γ z . To see this, note that if x ∈ Dz , y ∈ ∼ l∈N Blz and ρz (x, y) = γ z , then there is some l for which x ∈ Wlz , y ∈ / Wlz , so that x ∈ Clz ⊆ Blz . Proof of Theorem 4.2.7. The proof that 2 implies 1 is the same as before. Suppose now that 1 holds. For z ∈ Z, let T z be the tree of sequences of the form z (proj0 (Λ), . . . , proji (Λ)) for i ≤ n < N and Λ ∈ F̃ n . The map z 7→ T z is easily seen to be Borel. Define Tnz to be the n-th level of T z , as in the proof of Theorem 4.1.11, S z and let (Az,n Tn which intersect every element of k )k∈N be a sequence of subsets of z Tn , and for which every such subset appears at least once. Note that this can be done so that the map (z, n, k) 7→ Az,n k is Borel. Define now a family of Borel directed graphs Gn,k on D by (x, z)Gn,k (y, w) iff z = w, x 6= y, xE z y, and projn (ρz (x, y)) ∈ Az,n k . Suppose now that B ⊆ D is Gn,k -independent. Then projn (ρz (E z B z \ ∆(B z ))) does not contain any element of Tnz for all z ∈ Z. That is, for all z and all Λ ∈ Tnz , Λ 6⊂ projn (ρz (E z B z \ ∆(B z ))). Let (Λzk )k∈N be a sequence of elements of Tnz for which every element of Tnz appears at least once, and let K z be such that all elements of Tnz appear in {Λzk : k < K z }. As with Az,n k , this can be done in a uniformly Borel way. Then for all k ∈ N and all z ∈ Z 131 we have Λzk 6⊂ projn (ρz (E z B z \ ∆(B z ))), so by Proposition 4.2.8 we have a Borel map f : B × N → N so that for all z ∈ Z, k, l ∈ N, projn (ρz (E z (f z,k )−1 (l) \ ∆((f z,k )−1 (l)))) does not generate a group containing Λzk . Let g : N × Z → N<N be a Borel map such that g z maps N bijectively onto NK for all z ∈ Z and define h : B → N so that g(h(x, z), z) = (f (x, z, 0), . . . , f (x, z, K z − 1)). Note that h is Borel. Let Bl = h−1 (l) for l ∈ N. Then (Bl )l∈N is a cover of B by Borel sets, and we claim that for all z ∈ Z, l ∈ N, projn (ρz (E z Blz \ ∆(Blz ))) does not generate a group containing an element of Tnz . Indeed, every such group appears as Λzk for some k < K z , and x, y ∈ Blz =⇒ f (x, z, k) = f (y, z, k). In particular, f (x, z, k) = m ∈ N is constant for x ∈ Blz , and thus Blz ⊆ (f z,k )−1 (m). z This implies that for all z ∈ Z, l ∈ N there is some i so that projΓzi (ρz (E z Blz \ ∆(Blz ))) does not contain an element of F zi . Since the family F zi is a finite collection of finite sets, the property that projΓzi (ρz (E z Blz \ ∆(Blz ))) does not contain an element of F zi is co-analytic (as a subset of Z × N2 ). By the Number Uniformization Theorem, one can therefore refine the cover (Bl )l∈N of B to a cover (Bi,l )i,l∈N of B by Borel sets so z z that for all z ∈ Z, i, l ∈ N, projΓzi (ρz (E z Bi,l \ ∆(Bi,l ))) does not contain an element of F zi . It follows that if χB (G) ≤ ℵ0 then F is not an essential value of ρ. By Theorem 4.2.3, there is a sparse dense sequence of sets Sn,k ⊆ 2<N , and a continuous S homomorphism φ : 2ω → D from GS to G. We may choose S so that n,k Sn,k contains a sequence of every length. Then projZ ◦φ : 2ω → Z is a continuous homomorphism S from n,k GSn,k to ∆(Z), and it follows that projZ ◦φ is a constant. Let z ∈ Z be the value taken by projZ ◦φ. We therefore have a continuous homomorphism projX ◦φ : 2ω → Dz from GS to Gz . The rest follows as in the proof of Theorem 4.1.11. 4.3 Proofs of Lusin–Novikov dichotomies over quotients Let E, F be Borel equivalence relations on Polish spaces X, Y and P ⊆ X × Y be E × F -invariant. A uniformization of P is a set U ⊆ P whose sections Ux intersect Px in exactly one F -class. Let F ⊆ E be Borel equivalence relations on a Polish space X. We say F has index n in E if every E-class contains n F -classes, bounded finite index in E if there is a finite bound on the number of F -classes inside each E-class, and σ-bounded finite index in E if there is a cover of X by countably many Borel sets on which E has 132 bounded finite index over F . A transversal of E over F is a set B ⊆ X that intersects every E-class in exactly one F -class, and a (proper) quasi-transversal of E over F is a set B ⊆ X whose intersection with every E-class has non-empty intersection with a non-empty (proper) finite subset of the F -classes it contains. Proof of Theorem 4.1.6. To see that at most one of these alternatives hold, it suffices to show that there is no Borel E0,Λ -invariant transversal of E0 × I(n) over E0,Λ . Indeed, if πX , πY are as in alternative 2 and U is a Borel uniformization of P , then {y ∈ 2N ×n : ∃x(E0 ×I(n))y((πX (x), πY (y)) ∈ Ũ )} is a Borel E0,Λ -invariant transversal of E0 × I(n) over E0,Λ . Suppose now that B were such a transversal. Then B would be non-meagre, as countably many homeomorphic copies of B cover 2N ×n. It follows that for some i < n, A = {x ∈ 2N : (x, i) ∈ B} is non-meagre. By [Mila, Proposition 1.5], Λ ⊆ ρλ (E0 A \ ∆(A)), so in particular there are x, y ∈ A with xE0 y and ρλ (x, y)(i) 6= i (y, i) and (x, i), (y, i) ∈ B, a contradiction.  (as Λ is fixed-point-free). But then (x, i) E0,Λ To see that at least one of these alternatives hold, define E 0 = (E × I(Y ))P̃ , F 0 = (I(X) × F )P̃ , and Z = {z ∈ P̃ n : ∀i < j < n(z(i)(E 0 \ F 0 )z(j))}. For z, z 0 ∈ Z let z Ẽz 0 ⇐⇒ z(0)E 0 z 0 (0), so that Ẽ is a Borel equivalence relation on Z. There is a natural action of Sn on Z, namely the action (σz)(i) = z(σ −1 (i)) permuting the coordinates, and for all z Ẽz 0 there is a unique σ ∈ Sn such that (σz 0 )(i)F 0 z(i) for i < n (we are using here the fact that the sections of P have size exactly n). Let ρ(z, z 0 ) ∈ Sn be this σ, and note that ρ is a Borel cocycle on Ẽ. Case 1. Suppose that there is some fixed-point-free group Λ ≤ Sn , a redundant enumeration λ of Λ, and a continuous embedding π : 2N → Z of ρλ to ρ. By [Mila, Proposition 1.6] and a straightforward recursive construction, we may assume that Λ is minimal. Let xDy ⇐⇒ ∃i, j < n(π(x)(i)(1) = π(x0 )(j)(1)), and xF 00 y ⇐⇒ ∃i, j < n(π(x)(i)F 0 π(y)(j)). Note that F 00 is an equivalence relation on 2N , and that D ⊆ F 00 is closed. Claim 4.3.1. F 00 is meagre. Proof. By the Kuratowski–Ulam Theorem [Kec95, 8.41], it suffices to show that every F 00 -class C is meagre. Fix x ∈ C and for i, j < n let Ci,j = {y ∈ C : π(x)(i)F 0 π(y)(j)}. Suppose towards a contradiction that C is non-meagre, and fix i, j such that Ci,j is 133 non-meagre. By the proof of [dRM, Proposition 1.3] and the fact that λ is a redundant enumeration of Λ, for all λ ∈ Λ there are y, z ∈ Ci,j with yE0 z and ρλ (y, z) = λ. Since Λ is fixed-point-free, we may choose y, z ∈ Ci,j such that yE0 z and ρλ (y, z)(j) 6= j. But then π(y)(j) F0 π(z)(j) and π(y)(j)F 0 π(x)(i)F 0 π(z)(j), a contradiction. / By [Mila, Proposition 1.7], we may assume that D = ∆(2N ) and that x E0 x0 =⇒ 00 π(x) F π(x0 ). Let πX (x, i) = projX (π(x)(i)), πY (x, i) = projY (π(x, i)). Note that xE0 y ⇐⇒ π(x)Ẽπ(y) ⇐⇒ π(x)(i)E 0 π(y)(j) ⇐⇒ πX (x, i)EπX (y, j) for x, y ∈ 2N , i, j < n, and that (x, i)E0,Λ (y, j) ⇐⇒ xE0 y & ρλ (x, y)(j) = i ⇐⇒ π(x)(i)(E 0 ∩ F 0 )π(y)(j), so πX is a reduction of E0 × I(n) to E and πY is a homomorphism from E0,Λ to F . (y, j). If xE y, then ρ(π(x), π(y))(j) 6= i so  On the other hand, suppose that (x, i) E0,Λ 0 0 00 0  π(x)(i) Fπ(y)(j), and if x E0 y then x F y so π(x)(i) Fπ(y)(j). Thus πY is a reduction N of E0,Λ to F . Since D = ∆(2 ), it is easy to see that πY is injective. Suppose now that z(E0 × I(n))z 0 . Then (πX (z 0 ), πY (z 0 )) ∈ P̃ , and πX (z)EπX (z 0 ), so by E-invariance (πX (z), πY (z 0 )) ∈ P̃ . Therefore (πX × πY )(E0 × I(n)) ⊆ P̃ . Thus πX , πY witness alternative 2 in Theorem 4.1.6. Case 2. Suppose now that we are not in case 1. By Theorem 4.1.5 and [Mila, Proposition 1.4], for every subset Λ ⊆ Sn that generates a fixed-point-free subgroup there is a cover of Z by countably many Borel sets Bn for which Λ is not contained in ρ(ẼBn \ ∆(Bn )). Since there are only finitely many such sets Λ, we may find a cover Bn of Z by Borel sets satisfying that ρ(ẼBn ) does not generate a fixedpoint-free group, i.e., such that ρ(ẼBn ) has a fixed point when acting naturally on n. For all n, let in be such a fixed point and let An = projin (Bn ). Then An ⊆ P̃ is analytic, its sections intersect at most one F -class, and this F -class is invariant under E. By [dRM, Proposition 2.1], we may extend An to a Borel E × F -invariant set A0n ⊆ P̃ with this property. By [dRM, Proposition 3.4] the set [A0n ]E 0 is Borel, so S S A = n (A0n \ k<n [A0k ]E 0 ) is Borel, E × F -invariant, and its sections contain at most one F -class. To see that A has non-empty sections, note that the sets Bn cover Z and that for all x ∈ X there is some z ∈ Z with xEz(0)(0). 134 The proof of Theorem 4.1.8 is identical, except we show that when Λ is transitive then there is no Borel E0,Λ -invariant proper quasi-transversal of E0 over E0,Λ , and in case 2 we simply consider the projection proj0 (Bn ). Proof of Theorem 4.1.7. That these are mutually exclusive follows from [Mila, Proposition 1.5], as in the proof of Theorem 4.1.6. To see that at most one of them hold, we argue by induction on n. By [dRM, Theorem 2.12], the set A of x ∈ X/E for which Px contains exactly n points is Borel. S Indeed, fix Borel maps φn : X → Y so that Px = n [φn (x)]F for x ∈ X. Then x ∈ à ⇐⇒ ∃k0 , . . . , kn−1 ∀i < j < n(φki (x)F φkj (x)). Now we apply Theorem 4.1.6 to the restriction of E, P to Ã, and the inductive hypothesis to their restrictions to X \ Ã. Again, the proof of Theorem 4.1.9 is the same. We now consider the case of σ-bounded finite index. Proof of Proposition 4.1.10. Clearly 3 =⇒ 1 =⇒ 2. To see that 2 =⇒ 1, we apply [dRM, Proposition 2.1] to extend each P̃n to E × F -invariant sets with the same property. To see that 1 =⇒ 3 when F is strongly idealistic, we apply [dRM, Theorem 2.12] as in the proof of Theorem 4.1.7. Proof of Theorem 4.1.12. To see that at most one of these alternatives hold, it suffices to show (as in the proof of Theorem 4.1.6) that there is no Borel E0,ρ -invariant transversal of E0 × I(N) over E0,ρ , where Λi ≤ Γ̃αi is a coherent sequence of fixedpoint-free subgroups, λ is a redundant enumeration of Λ, and ρ : E0 → Γ̃α∞ is a Borel cocycle consistent with λ. So suppose that B were such a transversal. Then B would be non-meagre, as countably many homeomorphic copies of B would cover 2N × N, so there is some i ∈ N such that A = {x ∈ 2N : (x, i) ∈ B} is non-meagre. By Proposition 4.2.2 there is some γ ∈ Γ̃αi+1 so that γΛi+1 γ −1 ⊆ proji+1 (ρ(E0 A \ ∆(A))). Note that i < n(α, i + 1), so there is some j < n(α, i + 1) with γ(j) = i. Since Λi+1 is fixed-point-free, there is some λ ∈ Λi+1 for which λ(j) 6= j. Taking x, y ∈ A such (y, i) and (x, i), (y, i) ∈ B, a  that ρ(x, y) = γλγ −1 , we have ρ(x, y)(i) 6= i, so (x, i) E0,ρ contradiction. 135 To see that at least one of these alternatives hold, define E 0 = (E × I(Y ))P̃ , F 0 = (I(X) × F )P̃ , and let Z = {z ∈ (P̃n )α(n) : ∀i < j(z(i)(E 0 \ F 0 )z(j))}. Y n As in the proof of Theorem 4.1.6, we let z Ẽz 0 ⇐⇒ z(0)E 0 z 0 (0), and we note that for all z Ẽz 0 there is a unique element of S∞ , which we denote by ρ(z, z 0 ), such that (ρ(z, z 0 ) · z 0 )(i)F 0 z(i) for all i ∈ N. Here, S∞ acts on Z by permuting the coordinates, the choice of ρ(z, z 0 ) is unique because the sets Pn are pairwise disjoint, and for the same reason it is easy to see that ρ(z, z 0 ) ∈ Γ̃α∞ . This makes ρ : Ẽ → Γ̃α∞ a Borel cocycle. For i ∈ N, let F i denote the collection of subsets of Sα(i) that generate fixed-point-free subgroups of Sα(i) . Case 1. Suppose that F is not an essential value of ρ, and fix Borel sets Bi,k covering Z so that no element of F i is contained in projSα(i) (ρ(ẼBi,k \ ∆(Bi,k ))), i.e., projSα(i) (ρ(ẼBi,k \ ∆(Bi,k ))) has a fixed point j(i, k) ∈ α(i) for all i, k ∈ N. Thus, Ai,k = projn(α,i−1)+j(i,k) (Bi,k ) is an analytic set in P̃i whose sections intersect at most one F -class, and for which this F -class is invariant under E. By [dRM, Proposition 2.1] we can extend Ai,k to a Borel E × F -invariant set A0i,k ⊆ P̃i with the same property, and by [dRM, Proposition 3.4] the set [A0i,k ]E 0 P̃i is Borel. It follows that [A0i,k ]E 0 is Borel: It is clearly analytic, and it is co-analytic because x ∈ [A0i,k ]E 0 ⇐⇒ ∀y(yE 0 x & y ∈ P̃i =⇒ y ∈ [A0i,k ]E 0 P̃i ). It follows as in the proof of Theorem 4.1.6 that n (A0n \ k<n [A0k ]E 0 ) is a Borel E × F invariant uniformization of P̃ , and hence its quotient gives a Borel uniformization of P. S S Case 2. Suppose now that F is an essential value of ρ. By Theorem 4.1.11, there is a coherent sequence Λi ∈ F̃ i , a redundant enumeration λ of Λ, a Borel cocycle ρ : E0 → Γ̃α∞ consistent with λ, and a continuous embedding π : 2N → Z of ρ into ρ. Let xDk y ⇐⇒ ∃i, j < n(α, k)(π(x)(i)(1) = π(y)(j)(1)), and xF 00 y ⇐⇒ ∃i, j(π(x)(i)F 0 π(y)(j)). Note that F 00 is an equivalence relation on 2N , and that each Dk ⊆ F 00 is closed. Claim 4.3.2. F 00 is meagre. Proof. By the Kuratowski–Ulam Theorem, it suffices to show that every F 00 -class C is meagre. Fix x ∈ C, and for i, j let Ci,j = {y ∈ C : π(x)(i)F 0 π(y)(j)}. Suppose towards a contradiction that C is non-meagre, and fix i, j such that Ci,j is non-meagre. Let k be 136 such that i, j < n(α, k − 1), and fix s ∈ 2<N with |s| ≥ n(α, k) so that Ci,j is comeagre in Ns . Since Γ̃αk is finite, there is some γ ∈ Γ̃αk such that projk (ρ(s_ x, 0|s|_ x)) = γ for a non-meagre set A of x ∈ 2N . Let B = {0|s|_ x : x ∈ A & s_ x ∈ Ci,j } and note that B is non-meagre. By the proof of [dRM, Proposition 1.3], the fact that λ is a redundant enumeration of Λ, and the fact that ρ is consistent with λ, for all λ ∈ Λk there are y, z ∈ B with yE0 z and projk (ρ(y, z)) = λ. Since Λk has no fixed points, it follows that there are y, z ∈ Ci,j so that yE0 z and projk (ρ(y, z))(j) 6= j. But then π(y)(j) F0 π(z)(j) and π(y)(j)F 0 π(x)(i)F 0 π(z)(j), a contradiction. / By Proposition 4.2.1, we may assume that x(k) 6= y(k) =⇒ x D E0 y =⇒ k y and that x 00 π(x) Fπ(y). Let πX (x, i) = projX (π(x)(i)), πY (x, i) = projY (π(x)(i)). Note that xE0 y ⇐⇒ π(x)Ẽπ(y) ⇐⇒ π(x)(i)E 0 π(y)(j) ⇐⇒ πX (x, i)EπX (y, j) for x, y ∈ 2N , i, j ∈ N, and that (x, i)E0,ρ (y, j) ⇐⇒ xE0 y & ρ(x, y)(j) = i ⇐⇒ π(x)(i)(E 0 ∩ F 0 )π(y)(j), so πX is a reduction of E0 × I(N) to E and πY is a homomorphism from E0,ρ to F . (y, j). If xE0 y, then ρ(π(x), π(y))(j) 6= i so  On the other hand, suppose that (x, i) E0,ρ 00 π(x)(i) F0 π(y)(j), and if x E0 y then x F y so π(x)(i) F0 π(y)(j). Thus πY is a reduction from E0,ρ to F . Since x(k) 6= y(k) =⇒ x D k y, it is easy to see that πY is injective S when restricted to i∈N N0i × (n(α, i) \ n(α, i − 1)). Suppose now that z(E0 × I(N))z 0 . Then (πX (z 0 ), πY (z 0 )) ∈ P̃ , and πX (z)EπX (z 0 ), so by E-invariance (πX (z), πY (z 0 )) ∈ P̃ . Therefore (πX × πY )(E0 × I(N)) ⊆ P̃ . Thus πX , πY witness alternative 2 in Theorem 4.1.12. Proof of Theorem 4.1.13. That 1 and 2(a) are incompatible follows from (the proof of) Theorem 4.1.6. The same holds for 1 and 2(b) by Theorem 4.1.12. We may assume that for every x ∈ X, the sections (Pn )x are pairwise disjoint. By (the proof of) Proposition 4.1.10, we may also assume that the sections of each Pn are either empty or have size exactly n ≥ 2. Note that by [dRM, Theorem 2.12], the map x 7→ Ax ∈ 2N given by n ∈ Ax ⇐⇒ (Pn )x 6= ∅ is Borel. For A ∈ 2N , let X A = {x : A = Ax }. If A ∈ 2N is finite, we restrict to X A and apply Theorem 4.1.6. As there are only countably many such A, we may assume that 2(a) does not hold for any of them and hence that X A = ∅ for finite A. 137 For infinite A ∈ 2N , let β A ∈ NN be its increasing enumeration and define αA (n) ≥ 2 A to be the size of the non-empty sections of Pβ A (n) . Define Z A , Ẽ A , ρA , Γα = ΓA , F A as in the proof of Theorem 4.1.12 restricted to X A . It is not hard to verify that these definitions are uniformly Borel, and hence we may apply Theorem 4.2.7 to Z, Ẽ, ρ, Γ, F . Case 1. If F is not an essential value of ρ, let Bi,k be Borel sets witnessing this. Then A A for all i, k ∈ N and infinite A ∈ 2N , the elements of projΓAi (ρA (Ẽ A Bi,k \ ∆(Bi,k ))) fix some point in αA (i). By the Number Uniformization Theorem, there is a Borel assignment j A (i, k) of such a point. Define n(αA , i) as in the non-parametrized case, and note that (A, i) 7→ n(αA , i) is Borel. For i, k ∈ N, let A Ci,k (w, A) ⇐⇒ ∃z ∈ Bi,k (z(n(αA , i − 1) + j(i, k)) = w). A Then Ci,k ⊆ P̃ is an analytic set satisfying: for all A ∈ 2N Ci,k ⊆ P̃β A (i) , its sections intersect at most one F -class, and this F -class is invariant under E. By [dRM, 0 Proposition 2.1] we may extend Ci,k to E × F × ∆(2N )-invariant Borel sets Ci,k with 0 the same property. Note that the sets Di,j = projX×Y (Ci,k ) are Borel, as the vertical 0 sections of Ci,k are either empty or singletons, and these are E ×F -invariant sets whose sections contain at most one F -class. Note that [Di,j ]E 0 is Borel: It is analytic, and x∈ / [Di,j ]E 0 iff there are A ∈ 2N , y0 , . . . , yαA (i)−1 such that A = Ax and yl ∈ P̃β A (i) \ Di,j for l < αA (i). As in the proof of Theorem 4.1.6, we get a Borel uniformization of P . Case 2. If F is an essential value of ρ, then by Theorem 4.2.7 and the proof of Theorem 4.1.6 we see that 2(b) holds. 138 Chapter 5 DESCRIPTIVE DICHOTOMY THEOREMS Michael S. Wolman 5.1 Introduction There are many dichotomy theorems in descriptive set theory, that is, theorems showing that if a given structure is not “simple” then it contains a copy of a canonical “complicated” object. Perhaps the first example of such a dichotomy is Cantor’s Perfect Set Theorem, which states that a closed set in a Polish space is either countable, or contains a homeomorphic copy of the Cantor space. This was later generalized to analytic sets, and then to various other types of structures, such as equivalence relations and graphs. We recall now some of the most important dichotomy theorems for equivalence relations, namely Silver’s Theorem and the Harrington–Kechris–Louveau Theorem, which we also call the E0 dichotomy. If E is an equivalence relation on X and F is an equivalence relation on Y , a reduction from E to F is a map f : X → Y such that xEx0 ⇐⇒ f (x)F f (x0 ). If X, Y are Polish, we write E ≤B F if there is a Borel reduction from E to F , and E vc F if there is a continuous embedding (i.e. injective reduction) from E to F . We let ∆(X) denote the equality relation on a set X, and say E is smooth if E ≤B ∆(X) for a Polish space X. Finally, we write E0 for the “eventual equality” relation on 2N : xE0 y ⇐⇒ ∃n∀k ≥ n(xk = yk ). Theorem 5.1.1 (Silver). Let E be a co-analytic equivalence relation on a Polish space. Exactly one of the following holds: 1. E has countably many equivalence classes. 2. ∆(2N ) vc E. Theorem 5.1.2 (The E0 dichotomy, Harrington–Kechris–Louveau). Let E be a Borel equivalence relation on a Polish space. Exactly one of the following holds: 1. E is smooth. 139 2. E0 vc E. We note that the E0 dichotomy extends prior results of Glimm and Effros, who proved this in the case that E is induced by a continuous action of a locally compact Polish group (and in particular when E is countable). For graphs, we have the Kechris–Solecki–Todorčević Theorem, also referred to as the G0 dichotomy. Given directed graphs G, H on vertex sets X, Y , a homomorphism from G to H is a map f : X → Y such that xGx0 =⇒ f (x)Hf (x0 ). If G is a graph on a Polish space X, we say G has countable Borel chromatic number, written χB (G) ≤ ℵ0 , if there is a Borel homomorphism from G to the complete graph on N (where N has the discrete topology). Put another way, χB (G) ≤ ℵ0 if there is a cover of X by countably many Borel G-independent sets. To define the canonical “complicated” graph G0 , we need a few definitions. We say a set S ⊆ 2<N is sparse if it contains at most one sequence of every length, and dense if for all t ∈ 2<N there is some s ∈ S for which t ⊆ s. For s ∈ 2<N , let Gs denote the directed graph on 2N given by Gs = {(s_ 0_ x, s_ 1_ x) : x ∈ 2N }, where _ denotes concatenation of sequences and we identify i ∈ 2 with the sequence S (i) ∈ 21 . Let GS = s∈S Gs for S ⊆ 2<N , and write G0 to denote the graph GS for some fixed sparse dense set S. Theorem 5.1.3 (The G0 dichotomy, Kechris–Solecki–Todorčević). Let G be a directed analytic graph on a Polish space. Exactly one of the following holds: 1. χB (G) ≤ ℵ0 . 2. There is a continuous homomorphism of G0 into G. The graph G0 depends on the choice of S, however all such graphs are acyclic Borel graphs of uncountable Borel chromatic number, and GS embeds continuously as a subgraph of GS 0 for all such choices of S, S 0 . This was subsequently generalized to (directed) hypergraphs by Lecomte. The original proofs of all of these dichotomies used techniques of effective descriptive set theory. Miller later found a classical proof of the G0 dichotomy, and used this to give a classical proof of Silver’s Theorem and the E0 dichotomy for countable Borel 140 equivalence relations. Miller also proved various generalizations of the G0 dichotomy, and used these generalizations to give a classical proof of the E0 dichotomy in full. Notably, Miller showed that many descriptive set theoretic dichotomies follow from appropriate graph-theoretic dichotomies and simple Mycielski-style Baire category arguments. We refer the reader to [Mil12] for a survey of these results, as well as many other examples. As noted above, Miller’s proof of the E0 dichotomy used a new generalization of the G0 dichotomy, and it has remained an open problem since then whether there is a direct proof of the E0 dichotomy from the G0 dichotomy. We show that this is indeed the case. Theorem 5.1.4. There is a proof of the E0 dichotomy directly from the G0 dichotomy. We give a proof of this in Section 5.2. The remainder of this chapter explores other constructions related to these graphtheoretic dichotomies, which we summarize below. In what follows, we assume all graphs are directed, though all of our results and proofs apply identically to undirected graphs. We first consider dichotomies of Miller for sequences of graphs. If (Gn )n∈N is a sequence of graphs on a Polish space X, we write χB ((Gn )) ≤ ℵ0 if there is a cover of X by countably many Borel sets so that each of these sets is Gn -independent for some n ∈ N. If (Gn )n∈N , (Hn )n∈N are sequences of graphs on X, Y respectively, we say a map f : X → Y is a homomorphism from (Gn ) to (Hn ) if f is a homomorphism from Gn to Hn for all n ∈ N. Let S be a collection of subsets of 2<N . We say S is sparse if if every element of S is dense. S S is sparse, and dense Theorem 5.1.5 (G0 for sequences, [Mil12, Theorem 21]). Let (Sn )n∈N be a sparse and dense sequence of subsets of 2<N . For any sequence (Gn )n∈N of analytic graphs on a Polish space, exactly one of the following holds: 1. χB ((Gn )) ≤ ℵ0 . 2. There is a continuous homomorphism from (GSn )n∈N to (Gn )n∈N . We remark that this follows from the ℵ0 -dimensional G0 dichotomy (the generalization of the G0 dichotomy to ℵ0 -dimensional hypergraphs), as noted in [Mil12]. 141 Theorem 5.1.6 (G0 for sequences-of-sequences, [Mil22, Theorem 1]). Let (Sn )n∈N be a sparse and dense sequence of subsets of 2<N . For any doubly-indexed sequence (Gi,j )i,j∈N of analytic graphs on a Polish space X that is increasing in the second coordinate, i.e., Gi,j ⊆ Gi,j+1 , exactly one of the following holds: 1. There is a cover of X by a sequence of Borel sets (Bi )i∈N such that for all i ∈ N, χB ((Gi,j Bi )j∈N ) ≤ ℵ0 . 2. There is a map f : N → N and a continuous homomorphism from (GSn )n∈N to (Gn,f (n) )n∈N . We show in Section 5.3 that Theorem 5.1.6 follows from Theorem 5.1.5, and in particular that Theorem 5.1.7. The G0 dichotomy for doubly-indexed sequences of graphs follows from the ℵ0 -dimensional G0 dichotomy. Our proof uses in a crucial way the compactness of 2N , and hence generalizes immediately to d-dimensional hypergraphs for d finite. We show that compactness is in some sense necessary in this result, by showing that it fails in the ℵ0 -dimensional case. Theorem 5.1.8. The d-dimensional analogue of the G0 dichotomy for doubly-indexed sequences of graphs is true for d ∈ N, and false for d = ℵ0 . We also prove a dichotomy for triply-indexed sequences of graphs. Suppose now that G0 , . . . , Gn−1 is a finite sequence of graphs on a Polish space X, S and let G = i<n Gi . It is easy to see that χB (G) ≤ ℵ0 iff χB (Gi ) ≤ ℵ0 for all i < n. Thus, by the G0 dichotomy, if all of these graphs are analytic and there is a continuous homomorphism of G0 into G, then there is a continuous homomorphism of G0 into Gi for some i < n. In Section 5.4, we consider the following question: Given a continuous homomorphism of G0 into G, can we show that there must be a continuous homomorphism of G0 into some Gi without applying the G0 dichotomy? We show that this is indeed the case: Theorem 5.1.9. Let G0 , . . . , Gn−1 be a sequence of Baire measurable graphs on 2N S for which G0 = i<n Gi . Then one can construct a continuous embedding of G0 into Gi for some i < n, without applying the G0 dichotomy. 142 Our proof uses a Mycielski-style Baire category construction, and gives a concrete combinatorial condition that specifies which of the graphs Gi we will embed G0 into. This has potential applications to more complicated situations, where one cannot simply fix in advance some graphs to which we apply the G0 dichotomy. For example, this idea is used in the proofs of Proposition 4.2.1 and Theorem 4.1.11 where we recursively construct a graph-theoretic embedding, and at every step of the construction we have to choose one of finitely many subgraphs to continue to embed into in a way that depends on our previous choices. Acknowledgements. Research partially supported by NSF Grant DMS-1950475. We would like to thank Alexander Kechris for their guidance and suggestions, Ben Miller for their helpful comments and kind words, and Esther Nam for the encouragement and support. 5.2 A proof of the E0 dichotomy from the G0 dichotomy Let E be a Borel equivalence relation on a Polish space X. Let Y = X × X \ E, and put a Polish topology on Y making the projections continuous. Define a directed Borel graph G on Y by setting (a0 , b0 )G(a1 , b1 ) ⇐⇒ a0 Eb1 . Claim 5.2.1. E is smooth iff χB (G) ≤ ℵ0 . Proof. Note that E is smooth iff it has a countable Borel separating family, i.e., a sequence of Borel sets Bi ⊆ X so that aEb ⇐⇒ ∀i(a ∈ Bi ⇐⇒ b ∈ Bi ). Suppose E is smooth, and let Bi be a countable Borel separating family for E. Then S Y = i (Bi × ∼Bi ) ∪ (∼Bi × Bi ), and these sets are G-independent. Conversely, suppose that Ai is a cover of Y by countably many G-independent Borel sets. Then the pairs (proj0 (Ai ), proj1 (Ai )) are E-independent, meaning that if a ∈ proj0 (Ai ), b ∈ proj1 (Ai ) then aEb. By a standard reflection argument (see e.g. [dRM, Proposition 2.1]), there is a Borel E-invariant set Bi ⊇ proj0 (Ai ) so that (Bi , proj1 (Ai )) is E-independent, and it is easy to verify that the sets Bi form a countable Borel separating family for E. / Remark 5.2.2. Ben Miller has pointed out that the graph G we define here was first considered (at least in the context of descriptive set theory) by Louveau, and Claim 5.2.1 was essentially proven by Lecomte (see [LM08], bottom of page 2). 143 Suppose now that E is not smooth. Let S ⊆ 2<ω be a dense set which contains exactly one sequence of every finite length n ∈ ω. By the directed G0 dichotomy and the previous claim, there is a continuous homomorphism ψ : 2ω → Y from GS to G. Let now H be the (undirected) graph given by xHz ⇐⇒ ∃y(xGS y & zGS y), and define xF y iff x, y are in the same connected component of H. Let E 0 be the pullback of E along proj0 ◦ψ, and let D be the pullback of ∆(X) along proj0 ◦ψ. Claim 5.2.3. E 0 is meagre and D is closed and nowhere dense. Proof. As proj0 ◦ψ is continuous and D ⊆ E 0 , it suffices to show that E 0 is meagre. By the Kuratowski–Ulam Theorem, it suffices to show that every E 0 -class is meagre. Let C be an E 0 -class, and suppose for the sake of contradiction that C is nonmeagre. Then by [KST99, Proposition 6.2], there are x, y ∈ C with xG0 y. If we write ψ(x) = (a0 , b0 ), ψ(y) = (a1 , b1 ), then we have a0 Ea1 (as x, y are in the same E 0 -class) and a0 Eb1 (as ψ is a directed homomorphism from G0 to G). It follows that a1 Eb1 and ψ(y) = (a1 , b1 ) ∈ Y = X × X \ E, a contradiction. / Claim 5.2.4. Let R ⊆ 2ω ×2ω be meagre and D ⊆ 2ω ×2ω be closed and nowhere dense. Then there is a continuous homomorphism φ : (G0 , ∼∆(2ω ), ∼E0 ) → (F, ∼D, ∼R). Apply these two claims to E 0 , D, in order to get a continuous homomorphism φ : (G0 , ∼∆(2ω ), ∼E0 ) → (F, ∼D, ∼E 0 ). Then proj0 ◦ψ ◦ φ : 2ω → X is a continuous embedding of E0 into E. Indeed, by definition of D this composition is injective. Moreover, proj0 ◦ψ is a reduction of E 0 into E, and φ is a reduction of E0 into E 0 . (To see this last part, note that H ⊆ E 0 , and thus F ⊆ E 0 : if xHz, then there is some y so that xG0 y and zG0 y, in which case proj0 (ψ(x))E proj1 (ψ(y))E proj0 (ψ(z)).) Thus it remains only to prove Claim 5.2.4. Proof of Claim 5.2.4. We define directed graphs GS on 2n by setting, for u, v ∈ 2n , uGS v iff there are s ∈ S ∩ 2<n , t ∈ 2n−1−|s| so that u = s_ 0_ t, v = s_ 1_ t. We define (undirected) graphs H on 2n by uHv ⇐⇒ ∃w(uGS w & vGS w), and let F be the equivalence relation on 2n given by the connected components of H. (We abuse notation here and let GS , H, F denote graphs on 2n for all n ≤ ω.) Note that ∀? ∈ {G0 , H, F }, u, v ∈ 2n , x ∈ 2≤ω (u ? v ⇐⇒ u_ x ? v _ x). (∗) 144 Subclaim 5.2.5. For any u, v ∈ 2n , there are some u0 , v 0 ∈ 2k , k < ω so that u_ u0 F v _ v 0 . Proof. It is a standard fact that for all u, v ∈ 2n , n < ω there is an undirected path from u to v in GS . (One can prove this easily by induction on n, using (∗) and the fact that S contains a sequence of every finite length.) We prove the subclaim for all u, v ∈ 2n , n < ω, by induction on the distance from u to v in GS . This is clearly true if the distance is 0, i.e., u = v. Suppose now that u, v are neighbours, i.e., uGS v or vGS u. By the symmetry of F , we may assume wlog that uGS v. As S is dense, there is some t ∈ 2<ω with v _ t ∈ S. But then u_ t_ 1GS v _ t_ 1 by (∗), and v _ t_ 0GS v _ t_ 1, so that u0 = t_ 1, v 0 = t_ 0 works. Finally, suppose that we have shown that this holds for all u, v of distance at most m in GS , for m ≥ 1, and let u, v be of distance m + 1 apart in GS . Let w be the last vertex before v on an undirected path from u to v of length m + 1. By assumption, there are u0 , v 0 ∈ 2k with u_ u0 F w_ v 0 . Note that w_ v 0 , v _ v 0 are adjacent in GS by (∗), so by assumption there are u00 , v 00 ∈ 2l with w_ v 0_ u00 F v _ v 0_ v 00 . By (∗) and the transitivity of F , u_ u0_ u00 F w_ v 0_ u00 F v _ v 0_ v 00 , so the pair u0_ u00 , v 0_ v 00 ∈ 2k+l satisfies the subclaim for u, v. // Let now Un be a decreasing sequence of dense open sets in 2ω × 2ω so that D ∩ U0 = ∅ T and R ∩ n Un = ∅. We will define recursively pairs of finite sequences (tn0 , tn1 ) ∈ S m m n <ω satisfying for all n < ω: m>0 2 × 2 and maps φn : 2 → 2 1. φ0 (∅) = ∅; 2. φn+1 (u_ (i)) = φn (u)_ tni for u ∈ 2n , i ∈ 2; 3. Nφn (u)_ tn0 × Nφn (v)_ tn1 ⊆ Un for all u, v ∈ 2n ; and 4. if s ∈ S ∩ 2n , then φn (s)_ tn0 F φn (s)_ tn1 . (Here, Nt = {x ∈ 2ω : t ⊆ x} for t ∈ 2<ω .) Assuming this has been done, define φ(x) = n φn (xn) for x ∈ 2ω . Then φ : 2ω → 2ω is continuous, and by 2, 3 it is a homomorphism ∼∆(2ω ) → ∼D and ∼E0 → ∼R. It is also easy to see by 2 that for any u, v ∈ 2n , x ∈ 2ω , there is some y ∈ 2ω so S 145 that φ(u_ x) = φn (u)_ y, φ(v _ x) = φn (v)_ y. By 2, 4, and (∗), it follows that φ is a homomorphism from G0 to F . It remains to define the sequences (tn0 , tn1 ) and maps φn . Suppose we have constructed these sequences and maps for all m < n. We define φn to be the unique map satisfying 1, 2. Because Un is dense and open, there are sequences w0 , w1 so that Nφn (u)_ w0 × Nφn (v)_ w1 ⊆ Un for all u, v ∈ 2n . (That this is true for a single pair u, v is immediate; we recursively apply this fact to all such pairs, extending w0 , w1 at each step, in order to find sequences that work for all u, v.) We may ensure that |w0 | = |w1 | > 0. Let now s ∈ S ∩ 2n . By the subclaim, there are w00 , w10 ∈ 2k , k < ω so that φn (s)_ w0_ w00 F φn (s)_ w1_ w10 . We let tni = wi_ wi0 , and note that 3, 4 are satisfied by this choice. / 5.3 Dichotomies for doubly-indexed sequences of graphs We begin by giving a proof of Theorem 5.1.6 from Theorem 5.1.5. Proof of Theorem 5.1.6 from Theorem 5.1.5. Define graphs Hi on NN × X by (f, x)Hi (g, y) ⇐⇒ f (i) = g(i) & xGi,f (i) y. This is a sequence of analytic digraphs to which we apply Theorem 5.1.5. Case 1: χB ((Hi )) ≤ ℵ0 . For B ⊆ NN × X, let B i,j = {(f, x) ∈ B : f (i) = j}. Note that if B is Hi -independent, then projX (B i,j ) is Gi,j -independent for all j. Now fix a cover of NN × X by Borel sets Bi,k such that Bi,k is Hi -independent for all i, k. Let i,j Ai,j,k = projX (Bi,k ), and note that each Ai,j,k is analytic and Gi,j -independent. Fix T S Gi,j -independent Borel sets A0i,j,k ⊇ Ai,j,k , and let Ai = j k A0i,j,k . Then alternative 1 of Theorem 5.1.6 holds for the sets Ai , so it remains to show that these sets cover X. Suppose for the sake of contradiction that x ∈ / i Ai . Then x ∈ / i j k Ai,j,k , so there is some f : N → N such that x ∈ / Ai,f (i),k for all i, k. By assumption there are i,f (i) i, k with (f, x) ∈ Bi,k , in which case x ∈ projX (Bi,k ) = Ai,f (i),k , a contradiction. S S T S Case 2: There is a continuous homomorphism φ : 2ω → X from (GSi )i to (Hi )i . Then projNN ◦φ : 2N → NN is continuous, so its image is compact and hence bounded (pointwise), say by f ∈ NN . We verify that f, φX = projX ◦φ satisfy the second alternative of Theorem 5.1.6: Let xGSi y and φ(x) = (g, φX (x)), φ(y) = (h, φX (y)) for some g, h ∈ NN . Then g(i) = h(i) ≤ f (i) and φX (x)Gi,g(i) φX (y). Since the graphs G are increasing in the second coordinate, φX (x)Gi,f (i) φX (y). 146 One can state and prove a d-dimensional analogue of Theorem 5.1.5 for d ≤ ℵ0 , using the ℵ0 -dimensional G0 dichotomy. When d is finite, the graphs live on the compact space dN , so an identical proof to the one above for the d = 2 case gives a d-dimensional analogue of Theorem 5.1.6 for d finite. We now give two examples to show that for Theorem 5.1.6 (a) the assumption that (Gi,j )i,j is increasing in the second coordinate cannot be dropped, and (b) the ℵ0 dimensional analogue of this theorem is false. Example 5.3.1 (Sequences that are not increasing in the second coordinate). Let (Si )i be a dense sequence subsets of 2<N such that Si contains only sequences of length at least i. For i ∈ N fix an enumeration tij of 2i and let Si,j = {s ∈ Si : tij ⊆ s}. Let Gi,j = GSi,j . This is a collection of Borel graphs such that for all i ∈ N, there are only finitely-many Gi,j . Note moreover that this collection of graphs is not increasing in the second coordinate (in fact the graphs (Gi,j )j for i fixed are pairwise disjoint). For any i, j<2i Gi,j = GSi , so if χB (Gi,j B) ≤ ℵ0 for all j < 2i then χB (GSi B) ≤ ℵ0 . Thus by [KST99, Proposition 6.2], if B is Baire measurable then it is meagre. It follows that alternative 1 of Theorem 5.1.6 fails for (Gi,j )i,j . S Suppose for the sake of contradiction that there is some f : N → N and a continuous homomorphism φ : 2N → 2N of (GSi )i to (Gi,f (i) )i . Let ui = tif (i) for i ∈ N, which is possible as f (i) < 2i . We claim that the image of φ is contained in Nui for all i, where Nt = {x ∈ 2N : t ⊆ x}. Otherwise, there is some open set U ⊆ 2ω and some ui 6= v ∈ 2i such that φ maps U into Nv . There are x, y ∈ U with xGSi y, so by assumption φ(x)Gi,f (i) φ(y), and hence φ(x), φ(y) ∈ Ntif (i) = Nui , a contradiction. It follows that u0 ⊆ u1 ⊆ u2 . . . , and if u = i ui ∈ 2N then φ is the constant function with value u, a contradiction. Thus alternative 2 of Theorem 5.1.6 fails for (Gi,j )i,j as well. S Example 5.3.2 (ℵ0 -dimensional graphs). We define the notion of density for subsets of N<N exactly as we did for subsets of 2<N . We define Gℵs 0 , GℵS0 analogously as well, e.g., by setting Gℵs 0 = {(s_ n_ x)n∈N : x ∈ NN } for s ∈ N<N . Let (Si )i be a dense sequence of subsets of N<N such that Si contains only sequences of length at least i. For i ∈ N, fix an enumeration tij of Ni , and let S Si,j = {s ∈ Si : tij ⊆ s}. Let Gi,j = l≤j GℵS0i,l . This is a doubly-indexed sequence of Borel ℵ0 -dimensional digraphs which is increasing in the second coordinate. 147 Suppose B is Baire measurable, i ∈ N, and χB (Gi,j B) ≤ ℵ0 for all j. The usual argument shows that B is meagre in Ntij for all j, and hence B is meagre. It follows that alternative 1 of Theorem 5.1.6 fails for (Gi,j )i,j . Suppose now that there was some f : N → N, a comeagre set C ⊆ NN , and a continuous homomorphism φ : C → NN of (GℵSi0 )i to (Gi,f (i) )i . Let Ai = {tij : j ≤ f (i)}. S We claim that the image of φ is contained in {Nt : t ∈ Ai } for all i, which can be seen just as in the previous example. Let T be the tree of sequences whose initial segments are all contained in i Ai . Then the image of φ is contained in [T ], a compact set, which is bounded (pointwise). But every bounded set is Gi,j -independent for all i, j, a contradiction. S The dichotomy theorems we have discussed thus far all admit an effective analogue (see for example [KST99, Theorem 6.4]). In these effective dichotomies, the “simple” case has an effective witness, whereas the “complex” case does not. We show that this is also the case for the function f in alternative 2 of Theorem 5.1.6. Example 5.3.3 ((In)effective bounds on f ). Fix a coding F 3 n 7→ Fn of the ∆11 points in NN , i.e., sets F ∈ Π11 (N), P ∈ Π11 (N × NN ), S ∈ Σ11 (N × NN ) so that if n ∈ F then Pn = Sn are singletons, which we denote {Fn }, and so that every ∆11 point in NN appears in this way (see e.g. [Mos09, Section 4D]). Let A(i, j) ⇐⇒ ∀n ≤ i(n ∈ F =⇒ Fn (i) < j). Then A ∈ Σ11 and for all i, Ai is an interval [j, ∞), but there is no f ∈ ∆11 with A(i, f (i)) for all i ∈ ω. Indeed, if f were such a function, then there would be i ∈ F with f = Fi , in which case we would have f (i) = Fi (i) and thus ¬A(i, f (i)), a contradiction. Let S ⊆ 2<N be dense and computable, and let xGi,j y ⇐⇒ xGS y & A(i, j). Then (Gi,j )i,j is a (uniformly) Σ11 family of digraphs for which alternative 2 holds in Theorem 5.1.6. However, if f, φ witness this, then A(i, f (i)) for all i ∈ ω, so f ∈ / ∆11 . Problem 5.3.4. Can we find such an example where (Gi,j )i,j is uniformly ∆11 ? We give also a possible generalization of Theorem 5.1.6 to triply-indexed sequences of graphs. Theorem 5.3.5. Let (Si,k )i,k be a dense, sparse family of subsets of 2<ω . Let (Gi,j,k )i,j,k be a triply-indexed sequence of analytic digraphs on a Polish space X which is increasing in the second coordinate, i.e., Gi,j,k ⊆ Gi,j+1,k . For i, j ∈ N, let Gi,j = (Gi,j,k )k∈ω . Then exactly one of the following holds: 148 1. There is a cover of X by countably-many Borel sets Bi such that χB (Gi,j Bi ) ≤ ℵ0 for all i, j ∈ N. 2. There is a map f : N → N and a continuous homomorphism φ : 2N → X from (GSi,k )i,k∈N to (Gi,f (i),k )i,k∈N . Proof sketch. We argue as in our proof of Theorem 5.1.6, but we apply Theorem 5.1.5 to the family of graphs (f, x)Hi,k (g, y) ⇐⇒ f (i) = g(i) & xGi,f (i),k y. Problem 5.3.6. Is there a generalization of this to e.g. families of analytic graphs indexed by well-founded trees that satisfy some sort of monotonicity condition? 5.4 A notion of largeness for subgraphs of G0 Fix a sparse dense set S ⊆ 2<N and write G0 = GS . Assume for convenience that S contains exactly one sequence of every finite length, and let sn ∈ S ∩ 2n . Let (Hi )i∈l be a finite sequence of Baire measurable graphs on 2N such that G0 ⊆ i∈l Hi . We will show that there is a continuous embedding of G0 into Hi for some i ∈ l, without applying the G0 dichotomy. S _ _ _ To begin, define An,i = {x ∈ 2N : (s_ n 0 x, sn 1 x) ∈ Hi }, and note that each An,i is Baire measurable. Let Si = {sn : An,i is non-meagre}, and let Ti ⊆ 2<N be the smallest tree containing Si . Note that S i∈l Si = S is dense, so [ i∈l S i∈l Ti = 2 [Ti ] = [ [ <N . Moreover, we have Ti ] = [2<N ] = 2N i∈l because this collection of trees is finite. It follows that some [Ti ] is non-meagre in 2N . We may assume wlog that [T0 ] is non-meagre, and since it is closed, that there is some t0 ∈ 2<N such that t0 ⊆ t =⇒ t ∈ T0 . In other words, S0 is dense below t0 , meaning that if t0 ⊆ t then there is some s ∈ S0 with t ⊆ s. Finally, for each sn ∈ S0 we fix sn ∈ 2<N such that An,0 is comeagre in Nsn , and we fix a decreasing sequence of open sets (Ukn )k∈N that are dense in Nsn and which satisfy T U0n = Nsn and An,0 ⊇ k Ukn . For B ⊆ 2N and t ∈ 2<N , write t_ B = {t_ x : x ∈ B}. We will now recursively construct sequences un ∈ 2<N and a map f : 2<N → 2<N satisfying: 149 (1) t0 ⊆ f (∅) and f (s_ i) = f (s)_ i_ un for all s ∈ 2n , i ∈ 2; (2) for all n ∈ N there is some m = m(n) such that f (sn ) = sm ∈ S0 ; (3) Nf (sn _ i_ t) ⊆ sm(n) _ i_ Uk m(n) for all n ∈ N, i ∈ 2, t ∈ 2k . Suppose this has been done, and set π(x) = n f (xn) for x ∈ 2N . By (1), π is a continuous embedding of 2N into itself. By (1) and (2), for all n ∈ N and x ∈ 2N there is some m and y ∈ 2N such that sm ∈ S0 and π(sn _ i_ x) = sm _ i_ y for i ∈ 2, and by T (3) y ∈ k Ukm ⊆ Am,0 , so that (π(sn _ 0_ x), π(sn _ 1_ x)) ∈ H0 and π is a continuous embedding of G0 into H0 . S We now construct the sequences un and the map f . First, we let f (∅) ∈ S0 be such that t0 ⊆ f (∅), which is possible as S0 is dense below t0 . Suppose now that we have constructed ui , i < n and f : 2<n → 2<N satisfying (1)-(3) on 2<n . We begin by finding a sequence v ∈ 2<N such that sm(n−1) ⊆ v and for all m(k) k < n − 1, t ∈ 2n−k−2 , i, j ∈ 2 we have Nf (sk _ i_ t)_ j _ v ⊆ sm(k) _ i_ Un−k . To do so, it suffices to show that for any particular such k, t, i, j and any v 0 ∈ 2<N there is some v ⊇ v 0 which satisfies this condition, as we may then recursively extend sm(n−1) to define a v which works for all such quadruples. Given k, t, i, j, v 0 , this is always possible m(k) as Un−k is open and dense in Nsm(k) , and sm(k) _ i_ sm(k) ⊆ f (sk _ i_ t) by (3). Next, let t ∈ 2n−1 , j ∈ 2 be such that sn = t_ j. Since t0 ⊆ f (t) and S0 is dense below t0 , there is some u ⊇ v and m ∈ N such that f (t)_ j _ u = sm ∈ S0 . We then set un = u and m(n) = m, and define f (s_ i) = f (s)_ i_ un for all s ∈ 2n−1 , i ∈ 2. It is clear that f : 2≤n → 2<N satisfies (1) and (2) on 2≤n . To show that (3) is satisfied, m(n−1) note that f (sn−1 _ i) = f (sn−1 )_ i_ un ⊇ sm(n−1) _ i_ sm(n−1) and U0 = Nsm(n−1) , n−k−2 _ _ _ so (3) holds for sn−1 . For k < n − 1, t ∈ 2 , i, j ∈ 2, we have f (sk i t j) ⊇ f (sk _ i_ t)_ j _ v, so (3) holds for this quadruple by our choice of v. 150 Appendix A DITZEN’S EFFECTIVE VERSION OF NADKARNI’S THEOREM Alexander S. Kechris and Michael S. Wolman A.1 Introduction In effective descriptive set theory one often considers the following type of question: Suppose we are given a (lightface) ∆11 structure R on the Baire space N (like, e.g., an equivalence relation, graph, etc.) and a problem about R that admits a (classical) ∆11 (i.e., Borel) solution. Is there an effective, i.e., ∆11 , solution? For example, consider the case where R = E is a ∆11 equivalence relations which is smooth, i.e., admits a Borel function f : N → N such that xEy ⇐⇒ f (x) = f (y). Then it turns out that one can find such a function which is actually ∆11 . One often derives such results via an effective version of a dichotomy theorem, For instance, for the example of smoothness above we have the following classical version of the so-called General Glimm-Effros Dichotomy proved in [HKL90]. Below E0 is the equivalence relation on the Cantor space C given by xE0 y ⇐⇒ ∃m∀n ≥ m(x(n) = y(n)). Theorem A.1.1 (General Glimm-Effros Dichotomy, see [HKL90]). Let E be a Borel equivalence relation on the Baire space N . Then exactly one of the following holds: (i) E is smooth, i.e., admits a Borel function f : N → N such that xEy ⇐⇒ f (x) = f (y), (ii) There is a Borel injective function g : C → N such that xE0 y ⇐⇒ g(x)Eg(y). Now it turns out that the proof of this result in [HKL90] actually gives the following effective version: Theorem A.1.2 (Effective General Glimm-Effros Dichotomy, see [HKL90]). Let E be a ∆11 equivalence relation on the Baire space N . Then exactly one of the following holds: (i) E admits a ∆11 function f : N → N such that xEy ⇐⇒ f (x) = f (y). (ii) There is a Borel injective function g : C → N such that xE0 y ⇐⇒ g(x)Eg(y). 151 From this it is immediate that the smoothness of E is witnessed effectively as mentioned earlier. For more examples of such effectivity results see also the recent paper [Tho24]. In Ditzen’s unpublished PhD thesis [Dit92], it is shown that the notion of compressibility of a countable Borel equivalence relation (CBER) is effective, i.e., if a ∆11 CBER on the Baire space N is compressible, then it admits a ∆11 compression. This follows from an effective version of Nadkarni’s Theorem that we state below. First recall the following standard concepts. A CBER E on a standard Borel space X is a Borel equivalence relation all of whose classes are countable. A compression of E is an injective map f : X → X such that for each E-class C we have f (C) $ C. We say that E is compressible if it admits a Borel compression. Finally a Borel probability measure µ on X is invariant for E if for any Borel bijection f : X → X with f (x)Ex, ∀x, we have that f∗ µ = µ. We now have: Theorem A.1.3 (Nadkarni’s Theorem, see [Nad90] and [BK96]). Let E be a CBER on the Baire space N . Then exactly one of the following holds: (i) E is compressible, i.e., admits a Borel compression; (ii) E admits an invariant probability Borel measure. We include below Ditzen’s proof of the following effective version of Nadkarni’s Theorem: Theorem A.1.4 (Effective Nadkarni’s Theorem [Dit92]). Let E be a (lightface) ∆11 CBER on the Baire space N . Then exactly one of the following holds: (i) E admits a ∆11 compression; (ii) E admits an invariant probability Borel measure. As a consequence of the proof of the Effective Nadkarni Theorem we also obtain a proof of an effective version of the classical Ergodic Decomposition Theorem (see [Far62] and [Var63]). This provides a different proof, for the restricted case of invariant measures, of Ditzen’s Effective Ergodic Decomposition Theorem for quasi-invariant measures [Dit92]. First we recall the classical Ergodic Decomposition Theorem. For a CBER E on a standard Borel space X, we let INVE denote the space of E-invariant probability Borel 152 measures on X. We say µ ∈ INVE is ergodic for E if µ(A) ∈ {0, 1} for all E-invariant Borel sets A ⊆ X, and we let EINVE ⊆ INVE denote the space of E-ergodic invariant probability Borel measures on X. Theorem A.1.5 (Ergodic Decomposition Theorem, see [Far62] and [Var63]). Let E be a CBER on the Baire space N and suppose that INVE 6= ∅. Then EINVE 6= ∅ and there is a Borel surjection π : N → EINVE such that: (i) π is E-invariant; (ii) if Se = {x : π(x) = e}, for e ∈ EINVE , then e(Se ) = 1 and e is the unique E-ergodic invariant probability Borel measure on E|Se ; (iii) for any µ ∈ INVE , µ = π(x)dµ(x). R Nadkarni in [Nad90] noted that his proof of Theorem A.1.3 can be also used to give a proof of Theorem A.1.5. We will show below that this argument can also be effectivized. Let P (N ) denote the space of probability Borel measures on N . One can identify a probability Borel measure µ on N with the map ϕµ : N<N → [0, 1], ϕµ (s) = µ(Ns ), where Ns = {x ∈ N : s ⊆ x} (cf. [Kec95, 17.7]). In this way, one may view P (N ) as P <N the Π02 subset of [0, 1]N consisting of all ϕ satisfying ϕ(∅) = 1 and ϕ(s) = n ϕ(s_ n) for all s ∈ N<N . Via this identification, we will prove the following effective version of the Ergodic Decomposition Theorem: Theorem A.1.6 (Effective Ergodic Decomposition Theorem, see [Dit92]). Let E be a (lightface) ∆11 CBER on the Baire space N and suppose that INVE 6= ∅. Then <N EINVE 6= ∅, and there is a ∆11 E-invariant set Z ⊆ N and a ∆11 map π : Z → [0, 1]N such that: (i) E|(N \ Z) admits a ∆11 compression, i.e. there is a ∆11 injective map f : N \ Z → N \ Z such that f (C) $ C for every E-class C ⊆ N \ Z; (ii) π maps Z onto EINVE ; (iii) π is E-invariant; (iv) if Se = {x ∈ Z : π(x) = e}, for e ∈ EINVE , then e(Se ) = 1 and e is the unique E-ergodic invariant probability Borel measure on E|Se ; (v) for any µ ∈ INVE , µ = π(x)dµ(x). R 153 In Section A.4, we will show that there is a ∆11 CBER E on N that admits an invariant probability Borel measure but does not admit a ∆11 invariant probability measure. It follows that we cannot, in general, make the map π from Theorem A.1.6 total, because if we could then E would admit a ∆11 invariant probability measure. A.2 A representation of ∆11 equivalence relations In this section we will prove a representation of ∆11 CBER that is needed for the proof of Theorem A.1.4. It can be viewed as a strengthening and effective refinement of the Feldman-Moore Theorem, which asserts that every CBER is obtained from a Borel action of a countable group. Below we use the following terminology: Definition A.2.1. A sequence (An ) of ∆11 subsets of N is uniformly ∆11 if the relation A ⊆ N × N given by A(n.x) ⇐⇒ x ∈ An , is ∆11 . Similarly a sequence (fn ) of partial ∆11 functions fn : N → N (i.e., functions with ∆11 graph) is uniformly ∆11 if the partial function f : N × N → N given by f (n, x) = fn (x), is ∆11 . We also say that a countable collection of subsets of N is uniformly ∆11 if it admits a uniformly ∆11 enumeration. Similarly for a countable set of partial functions. Theorem A.2.2 ([Dit92, Section 2.2.1]). Let E be a ∆11 CBER on the Baire space N . Then (1) E is induced by a uniformly ∆11 sequence of (total) involutions, i.e., there is a such a sequence (fn ) with xEy ⇐⇒ ∃n(fn (x) = y). (2) There is a Polish 0-dimensional topology τ on N , extending the standard topology, and a uniformly ∆11 countable Boolean algebra U of clopen sets in τ , which is a basis for τ and is closed under the group Γ generated by (fn ). (3) There is a complete compatible metric d for τ such that for every U ∈ U and k > 0, S there is a uniformly ∆11 , pairwise disjoint, sequence (Unk ) with Unk ∈ U , U = n Unk and diamd (Unk ) < k1 , and such that moreover the sequence (Unk ) is uniformly ∆11 in U, k, n. Proof. For (1): This follows immediately from the usual proof of the Feldman-Moore Theorem (see [FM77] or [Slu, Section 1.2]). So fix below such a sequence (fn ) and consider the corresponding ∆11 action of the group Γ. 154 For (2), (3): We will first find a topology τ as in (2), which has a uniformly ∆11 countable basis B of clopen sets closed under the Γ-action, because we can then take U to be the Boolean algebra generated by B. For (3) we will find a complete compatible ∆11 metric d for τ (i.e., d : N 2 → R is ∆11 ). Then if (Un ) is a uniformly ∆11 enumeration of U, we have that A(k, n) ⇐⇒ diamd (Un ) < 1 k+1 is Π11 and ∀x ∈ N ∀k∃n(n ∈ Ak & x ∈ Un ), where Ak = {n : (k, n) ∈ A}. So, by the Number Uniformization Theorem for Π11 , there is a ∆11 function f : N ×N → N such that ∀x ∈ N ∀k(f (x, k) ∈ Ak & x ∈ Uf (x,k) ). Since A0 (k, n) ⇐⇒ ∃x ∈ N (n = f (x, k)) is a Σ11 subset of A, let A00 be ∆11 such that A0 ⊆ A00 ⊆ A. Since [ N ×N= Un × {k}, (k,n)∈A00 we can find a uniformly ∆11 sequence (Xnk ) of sets in U, such that for all k > 0 the sequence (Xnk )n is a partition of N of sets with d-diameter less than k1 . Finally given any U ∈ U , let Unk = Xnk ∩ U . It thus remains to find τ, d with these properties. We will need first a couple of lemmas. Lemma A.2.3. Let A ⊆ N be ∆11 . Then there is a Polish 0-dimensional topology τA on N , which extends the standard topology, has a uniformly ∆11 countable basis consisting of clopen sets containing A, and has a complete compatible ∆11 metric dA . Proof. Let f : N → N be computable and let B ⊆ N be Π01 and such that f |B is injective and f (B) = A. Use f to move the (relative) topology of B to A and the standard metric of B to A. Do the same for N \ A and then take the direct sum of these topologies and metrics on A, N \ A to find τA , dA . Lemma A.2.4. Let A = (An ) be a uniformly ∆11 sequence of subsets of N . Then there is a Polish 0-dimensional topology τA on N , which extends the standard topology, has a uniformly ∆11 countable basis BA containing all the sets in A, and has a complete compatible ∆11 metric dA . 155 Proof. Consider τAn , dAn as in Lemma A.2.3. Then put τA = the topology generated by [ τAn . n Then by [Kec95, Lemma 13.3], τA is Polish (and contains the standard topology). A basis for τA consists of all sets of the form U1 ∩ U2 ∩ · · · ∩ Un , where Ui ∈ BAji , 1 ≤ I ≤ n, and so it is 0-dimensional with a uniformly ∆11 basis BA containing all the sets in A. Finally, as in the proof of [Kec95, Lemma 13.3] again, a complete compatible metric for τA is X dAn (x, y) dA (x, y) = 2−n−1 · . 1 + dAn (x, y) n Because of the uniformity in A of the proof of Lemma A.2.3 this metric is also ∆11 . We finally find τ, d. To do this we recursively define a sequence of Polish 0-dimensional topologies τ0 , τ1 , . . . on N , extending the standard topology, and uniformly ∆11 countable bases Bn for τn and complete compatible ∆11 metrics dn for τn , all uniformly in n as well, and such that Γ · Bn ⊆ Bn+1 . For n = 0, let τ0 , d0 , B0 be the standard topology, metric and basis for N . Given τn , dn , Bn , consider Γ·Bn and use Lemma A.2.4 to define τn+1 , Bn+1 ⊇ Γ·Bn , dn+1 . The uniformity in n is clear from the construction. Finally let τ be the topology generated by n τn . It is 0-dimensional, Polish, with basis the sets of the form U1 ∩ U2 ∩ · · · ∩ Un , S with Ui ∈ Bji , 1 ≤ i ≤ n, so this is a uniformly ∆11 countable basis B consisting of clopen sets. Also clearly for any γ ∈ Γ, γ · (U1 ∩ U2 ∩ · · · ∩ Un ) = γ · U1 ∩ γ · U2 ∩ · · · ∩ γ · Un , where γ · Ui ∈ Bji +1 , thus γ · (U1 ∩ U2 ∩ · · · ∩ Un ) ∈ B as well. Finally, as before, a complete compatible ∆11 metric for τ is d(x, y) = X n and the proof is complete. 2−n−1 · dn (x, y) 1 + dn (x, y) 156 A.3 Proof of Effective Nadkarni In this section we show, using the representation of ∆11 CBER constructed in Section A.2, that we can effectivize the proof of Nadkarni’s Theorem. Our proof follows the exposition in [Dit92, Section 2.2.3]; see also the presentations of the classical proof in [BK96] or [Slu]. The classical proof of Nadkarni’s Theorem proceeds as follows. Fix a CBER E on N . We first define a way to compare the “size” of sets. For Borel sets A, B ⊆ N we write A ∼B B if there is a Borel bijection g : A → B with xEg(x), ∀x ∈ A. We write A ≺B B if there is some B 0 ⊆ B with A ∼B B 0 and [B]E = [B \ B 0 ]E , and A ≈B nB if we can partition A into pieces A0 , . . . , An so that Ai ∼B B for i < n and An ≺B B. One thinks of A ≈B nB to mean that A is about n times the size of B. In particular, if A ≈B nB and µ is an E-invariant probability Borel measure, then nµ(B) ≤ µ(A) ≤ (n + 1)µ(B). Note that E is compressible iff N ≺B N . More generally, we say that A ⊆ N is compressible if A ≺B A, i.e., if the equivalence relation E|A is compressible. Next we construct a fundamental sequence for E, i.e., a decreasing sequence (Fn ) of Borel sets such that F0 = N and Fn+1 ∼B Fn \ Fn+1 . Each Fn is a complete section for E, and is a piece of N of “size” 2−n , in the sense that N ≈B 2n Fn and µ(Fn ) = 2−n for all E-invariant probability Borel measures µ. It follows that if A ≈B kFn then k2−n ≤ µ(A) ≤ (k + 1)2−n for any E-invariant probability Borel measure µ. We then use the relative size of A with respect to the Fn to approximate what the measure of A would be with respect to some E-invariant probability Borel measure. To F do this, we construct, for all m, a partition [A]E = n≤∞ QA,m of [A]E into E-invariant n A,m Borel pieces such that Q∞ admits a Borel compression and A ∩ QA,m ≈B n(Fm ∩ n A,m Qn ) for n < ∞. We define the fraction function [A/Fm ] by setting [A/Fm ](x) = n if x ∈ QA,m or if n = 0 & x ∈ / [A]E , and let the local measure function n [A/Fm ](x) m(A, x) = limm→∞ [N /Fm ](x) . We show that m(A, x) is well-defined modulo an Einvariant compressible set, meaning there is an E-invariant set C ⊆ N admitting a Borel compression and such that m(A, x) is well-defined when x ∈ / C. We also show F P that for any partition A = n An into Borel pieces we have m(A, x) = n m(An , x) modulo an E-invariant compressible set, and if A ∼ B then m(A, x) = m(B, x) modulo an E-invariant compressible set. Finally, we show that the local measure function m(·, x) defines an E-invariant probability Borel measure, for all x ∈ N \ C, where C ⊆ N is some E-invariant 157 compressible set. To see this, we fix a Borel action Γ y N of a countable group Γ inducing E, a zero-dimensional Polish topology τ on N extending the usual one in which the action Γ y N is continuous, a complete compatible metric d for τ and a countable Boolean algebra of clopen-in-τ sets closed under the Γ action forming a basis for τ , and satisfying additionally that for every U ∈ U and k > 0 there is S a pairwise disjoint sequence (Unk ) of sets in U with U = n Unk and diamd (Unk ) < k1 . For each U ∈ U, k > 0 we fix such a sequence. Since the countable union of Borel E-invariant compressible sets is itself a Borel E-invariant compressible set, it follows that there is an E-invariant compressible set C ⊆ N such that for x ∈ / C we have P m(U, x) = n m(Unk , x) for U ∈ U, k > 0, m(U ∪ V, x) = m(U, x) + m(V, x) for U, V ∈ U disjoint, and m(U, x) = m(γU, x) for U ∈ U , γ ∈ Γ. Using this, we show that for x ∈ / C there is an E-invariant probability Borel measure µ with µ(U ) = m(U, x) for U ∈ U . It follows that either C = N , in which case E is compressible, or E admits an invariant probability Borel measure. In order to prove the effective version of Nadkarni’s Theorem, we will show that the classical proof outlined above can be effectivized using the representation in Section A.2. For the remainder of this section, we fix a ∆11 CBER E on N and a uniformly ∆11 sequence of (total) involutions (γn ) inducing E, as in Theorem A.2.2(1). Moreover, we assume, without loss of generality, that E is aperiodic, meaning that every E-class is infinite, because if C ⊆ N were a finite E-class then the uniform measure on C would be an E-invariant probability Borel measure. (A) Comparing the “size” of sets. We begin by defining a way to compare the “size” of ∆11 sets. The notation we use is the same as the notation typically used for the equivalent classical notions (cf. [Slu, Definition 2.2.4, Section 2.3]), which we denoted with the subscript B above. In this paper, these notions will always refer to the effective definitions below. Definition A.3.1. Let A, B ⊆ N be ∆11 . (1) We write A ∼ B if there is a ∆11 bijection f : A → B and such that xEf (x), ∀x ∈ A. If f is such a function we write f : A ∼ B. (2) We write A  B if A ∼ B 0 for some ∆11 subset B 0 ⊆ B. If f is such a function we write f : A  B. 158 (3) We write A ≺ B if there is some f : A  B such that [B \ f (A)]E = [B]E . If f is such a function we write f : A ≺ B. (4) We say A admits a ∆11 compression if A ≺ A, and if f : A ≺ A then we call f a ∆11 compression of A. (5) We write A  nB if there are ∆11 sets Ai , i < n such that A = for i < n. Note that A  1B ⇐⇒ A  B. S i<n Ai and Ai  B (6) We write A ≺ nB if in the previous definition there is some i < n for which Ai ≺ B. Note that A ≺ 1B ⇐⇒ A ≺ B. (7) We write A  nB if there are pairwise disjoint ∆11 sets Bi ⊆ A, i < n such that Bi ∼ B. (8) We write A ≈ nB if there is a partition A = i<n Bi t R into ∆11 pieces such that Bi ∼ B and R ≺ B. In particular, A ≈ 0B ⇐⇒ A ≺ B. Note that A ≈ nB implies that A  nB and A ≺ (n + 1)B. F We also let H denote the set of all E-invariant ∆11 subsets C ⊆ N that admit a ∆11 compression. Lemma A.3.2. (1) Let A ⊆ N be ∆11 . If A ≺ A then [A]E ≺ [A]E . (2) Let (An ), (Bn ) be uniformly ∆11 families of E-invariant sets and let (fn ) be a S S uniformly ∆11 sequence of maps satisfying fn : An ≺ Bn . Then n An ≺ n Bn . The same holds when ≺ is replaced by  or ∼, or if these are sequences of pairwise disjoint but not necessarily E-invariant sets. (3) Let A, B, C ⊆ N be ∆11 . If A  nB and C  mB for some m ≤ n, then C  A. If additionally C ≺ mB then C ≺ A. Proof. (1) Let f : A ≺ A and let g(x) = f (x) for x ∈ A, g(x) = x for x ∈ [A]E \ A. Then g : [A]E ≺ [A]E . (2) For x ∈ S n Bn . S n An set f (x) = fn (x) where n is least with x ∈ An . Then f : S n An ≺ (3) Let Ai , i < n be pairwise disjoint ∆11 subsets of A, fi : Ai ∼ B for i < n, Cj , j < m be ∆11 sets covering C and gj : Cj  B for j < m. Define h(x) = fj−1 ◦ gj (x) for j least with x ∈ Cj . 159 Then h : C  A, and if gj : Cj ≺ B then, letting C 0 = Cj \ S k<j Ck , we have [A \ h(C)]E ⊇ ([A]E \ [B]E ) ∪ [B \ gj (C 0 )]E = ([A]E \ [B]E ) ∪ [B]E = [A]E , so f : C ≺ A. (B) Fundamental sequences. Definition A.3.3. A uniformly ∆11 fundamental sequence for E is a uniformly ∆11 decreasing sequence (Fn ) of sets and a uniformly ∆11 sequence (fn ) of maps such that F0 = N and fn : Fn+1 ∼ Fn \ Fn+1 for all n. Lemma A.3.4. Let X ⊆ N be a ∆11 set on which E|X is aperiodic. Then there is a partition X = A t B of X into ∆11 pieces such that A ∼ B. In particular, E|A, E|B are also aperiodic. Proof. Let < be a ∆11 linear order on N (for example the lexicographic order) and let x ∈ An ⇐⇒ x < γn x. Define recursively the sets Ãn = {x ∈ X ∩ An : x, γn x ∈ X \ [ (Ãi ∪ γi Ãi )}. i<n Let A = n Ãn and define f = n γn |Ãn : A → X. Because of the uniformity of this construction, A, f are ∆11 . It is easy to see that f is injective and that f (A) ∩ A = ∅, so in particular that f : A ∼ f (A). F S We claim that A ∪ f (A) omits at most one point from each E|X-class. To see this, let S x < y ∈ X and suppose that xEy. Let γn x = y. If x, y ∈ / i<n (Ãi ∪ γi Ãi ), then by definition we have x, y ∈ Ãn ∪ γn Ãn ⊆ A ∪ f (A). Now let T = X \ (A ∪ f (A)), Y = X ∩ [T ]E , Z = X \ [T ]E . Then T is a traversal of E|Y and f |(A ∩ Z) : A ∩ Z ∼ f (A) ∩ Z. Thus it remains to prove the lemma for E|Y . In this case, using T and the sequence (γn ), one can enumerate each E|Y -class, and since these are infinite we can take A (resp. B) to be the even (resp. odd) elements of this enumeration. Proposition A.3.5. There exists a uniformly ∆11 fundamental sequence for E. Proof. We construct the sequences recursively. Let F0 = N and recursively apply Lemma A.3.4 to get Fn+1 and fn : Fn+1 ∼ Fn \ Fn+1 . Uniformity of these sequences follows from the uniformity in the proof of Lemma A.3.4. 160 For the remainder of this section, we fix a uniformly ∆11 fundamental sequence (Fn ) for E. (C) Decompositions of ∆11 sets. Lemma A.3.6. Let A, B ⊆ N be ∆11 and let Z = [A]E ∩ [B]E . There is a partition Z = P t Q of Z into E-invariant uniformly ∆11 sets such that A ∩ P ≺ B ∩ P and B ∩ Q  A ∩ Q. Proof. Define recursively the sets An = {x ∈ A \ [ [ Ai : γn x ∈ B \ i<n Bi }, Bn = γn An . i<n Let à = n An , B̃ = n Bn and f = n γn |An . By the uniformity of this construction, Ã, B̃, f are all ∆11 , so that f : à ∼ B̃. If we set P = Z ∩ [B \ B̃]E , Q = Z \ P then it is easy to see that A ∩ P ⊆ Ã, B ∩ Q ⊆ B̃ and hence that f |(A ∩ P ) : A ∩ P ≺ B ∩ P and f −1 |(B ∩ Q) : B ∩ Q  A ∩ Q. S S S Proposition A.3.7. Let A, B ⊆ N be ∆11 and let Z = [A]E ∩ [B]E . There exists a F partition Z = n≤∞ Qn of Z into E-invariant ∆11 pieces such that A ∩ Qn ≈ n(B ∩ Qn ) for n < ∞ and Q∞ ∈ H . Proof. We recursively construct sequences of sets An , Bn , P̃n , Q̃n , fn , gn , B̃n , Qn , Rn , Bni , fni for i < n such that A ∩ Qn = i<n Bni t R for n < ∞, fni : Bni ∼ B ∩ Qn for i < n < ∞, and fn : Rn ≺ B ∩ Qn for n < ∞. F First we let A0 = A, B0 = B. We apply Lemma A.3.6 to these sets to get P̃0 , Q̃0 , f0 , g0 and B̃0 satisfying f0 : A0 ∩ P̃0 ≺ B0 ∩ P̃0 , g0 : B0 ∩ Q̃0  A0 ∩ Q̃0 , B̃0 = Im(g0 ). Define Q0 = P̃0 , R0 = A0 ∩ Q0 . Now let n > 0 and suppose we have already constructed Ak , Bk , P̃k , Q̃k , fk , gk , B̃k , Qk , Rk , Bki , fki for all i < k < n. Let An = (An−1 ∩ Q̃n−1 )\ B̃n−1 , Bn = B ∩ Q̃n−1 . Apply Lemma A.3.6 to An , Bn to get P̃n , Q̃n , fn , gn , B̃n such that fn : An ∩ P̃n ≺ Bn ∩ P̃n , gn : Bn ∩ Q̃n  An ∩ Q̃n , B̃n = Im(gn ). 161 Define Qn = Q̃n−1 \ Q̃n , Rn = An ∩ Qn , Bni = B̃i ∩ Qn , fni = (gi )−1 |Bni . By uniformity of this construction it is clear that these sequences are uniformly ∆11 . Additionally, A ∩ Qn ≈ n(B ∩ Qn ) for n < ∞. Now let Q∞ = Z \ n Qn = n Q̃n . The sets B̃n are pairwise disjoint and gn : B ∩ Q̃n ∼ n n B̃n for all n. Therefore, if we define B∞ = B̃n ∩ Q∞ , g∞ = gn |(B ∩ Q∞ ) and n,m m n −1 n n,m n g∞ = g∞ ◦ (g∞ ) , we have that the B∞ are pairwise disjoint and g∞ : B∞ ∼ S S m n n,n+1 B∞ . Let B∞ = n B∞ and g∞ = n g∞ . Then B∞ , g∞ are ∆11 and g∞ : B∞ ≺ 0 B∞ . Since [B∞ ]E = [B∞ ]E = [B ∩ Q∞ ]E = Q∞ , Q∞ admits a ∆11 compression by Lemma A.3.2(1). S T Notation A.3.8. For ∆11 sets A, B ⊆ N , we let QA,B n , n ≤ ∞ be the decomposition of [A]E ∩ [B]E constructed in Proposition A.3.7. (D) The fraction functions. Definition A.3.9. We associate to all ∆11 sets A, B ⊆ N a fraction function [A/B] : N → N defined by " #   n A (x) =  B 0 if x ∈ QA,B for some n ≤ ∞, n otherwise. Lemma A.3.10. Let A, A0 , A1 , A2 , B, S ⊆ N be ∆11 . (1) If xEy then [A/B](x) = [A/B](y). (2) If A0  A1 then there is some C ∈ H such that [A0 /B](x) ≤ [A1 /B](x) for x ∈ / C. (3) If A0 ∼ A1 then there is some C ∈ H such that [A0 /B](x) = [A1 /B](x) for x ∈ / C. (4) If S is E-invariant then there is some C ∈ H such that for x ∈ S \ C we have [A/B](x) = [(A ∩ S)/B](x). (5) If A0 , A1 are disjoint then there is some C ∈ H such that for x ∈ / C, [A0 /B] + [A1 /B] ≤ [(A0 ∪ A1 )/B] ≤ [A0 /B] + 1 + [A1 /B] + 1. (6) If A1 is an E-complete section then there is some C ∈ H such that for x ∈ / C, [A0 /A1 ][A1 /A2 ] ≤ [A0 /A2 ] < ([A0 /A1 ] + 1)([A1 /A2 ] + 1). (7) There is some C ∈ H such that [Fn /Fn+m ] = 2m holds for all m, n ∈ N, x ∈ / C. 162 (8) There is some C ∈ H such that for all x ∈ [A]E \ C we have [A/Fn ](x) → ∞. (9) The set Y = {x : [A0 /B](x) < [A1 /B](x)} is ∆11 and E-invariant and A0 ∩ Y  A1 ∩ Y . Proof. (1) This is clear, as the sets QA,B are E-invariant. n 0 ,B 1 ,B (2) Let Cn,m = QA ∩ QA for m < n. Then A0 ∩ Cn,m ≈ n(B ∩ Cn,m ) and n m A1 ∩ Cn,m ≈ m(B ∩ Cn,m ) so by Lemma A.3.2(3) and our assumption we have A0 ∩ Cn,m  A1 ∩ Cn,m ≺ A0 ∩ Cn,m . By Lemma A.3.2(2) and the uniformity of S the proofs of Proposition A.3.7 and Lemma A.3.2(3), C = m<n Cn,m ∈ H , and [A0 /B](x) ≤ [A1 /B](x) for x ∈ / C. (3) This follows from (2). (4) As in the proof of (2), it suffices to show that C = S ∩ QA,B ∩ QA∩S,B admits a ∆11 k l compression (in a uniform way) for k 6= l. But A ∩ C ≈ k(B ∩ C) and A ∩ C ≈ l(B ∩ C) by E-invariance of C, so by Lemma A.3.2(1),(3) C admits a ∆11 compression. 0 ,B 1 ,B 2 ,B (5) Let C = Ci,j,k = QA ∩ QA ∩ QA . Then (5) fails to hold exactly when i j k x ∈ Ci,j,k for k < i + j or k > i + 1 + j + 1. Therefore, as in the proof of (2), it suffices to show that Ci,j,k admits a ∆11 compression (in a uniform way) for such i, j, k. Now we know that A0 ∩ C ≈ i(B ∩ C), A1 ∩ C ≈ j(B ∩ C), A2 ∩ C ≈ k(B ∩ C) by E-invariance of C. If k < i + j then (A0 ∪ A1 ) ∩ C ≺ (i + j)(B ∩ C) and (since A0 , A1 are disjoint) we have (A0 ∩C)∪(A1 ∩C)  (i+j)(B ∩C). Thus by Lemma A.3.2(1),(3) C = [(A0 ∪ A1 ) ∩ C]E admits a ∆11 compression. On the other hand, if k > i + 1 + j + 1 then (A0 ∩ C) ∪ (A1 ∩ C) ≺ (i + 1 + j + 1)(B ∩ C) and (A0 ∪ A1 ) ∩ C  k(B ∩ C), so again C admits a ∆11 compression. 0 ,A1 1 ,A2 (6) If x ∈ / [A0 ]E ∪ [A2 ]E then this clearly holds. Thus if C = Ck,l,m = QA ∩ QA ∩ k l A0 ,A2 Qm then (6) fails to hold exactly when x ∈ Ck,l,m for m < kl or m ≥ (k + 1)(l + 1). Therefore, as in the proof of (2), it suffices to show that these sets admit a ∆11 compression (in a uniform way). Since A0 ∩ C ≈ k(A1 ∩ C) and A1 ∩ C ≈ l(A2 ∩ C) we have that A0 ∩ C  kl(A2 ∩ C). Also, A0 ∩ C ≈ m(A2 ∩ C), so if kl > m then by Lemma A.3.2(1),(3) we are done. On the other hand, if m ≥ (k + 1)(l + 1) then A0 ∩ C  (k + 1)(l + 1)(A2 ∩ C), and since A1 ∩ C ≈ l(A2 ∩ C) one easily sees that A0 ∩ C  (k + 1)(A1 ∩ C). Thus by Lemma A.3.2(1),(3) we are done. 163 (7) Again it suffices to show that Qk n n+m admits a ∆11 compression in a uniform way for k 6= 2m . When k = ∞ this is clear. Otherwise, one easily sees by definition of the fundamental sequence that Fn ≈ 2m Fn+m , and moreover there is a uniformly ∆11 F ,F F ,F sequence of witnesses to this. It follows that Fn ∩ Qk n n+m ≈ 2m (Fn+m ∩ Qk n n+m ) F ,F F ,F and Fn ∩ Qk n n+m ≈ k(Fn+m ∩ Qk n n+m ), so when k 6= 2m this follows from Lemma A.3.2(1),(3). F ,F (8) Let C0 be the set constructed in (7), C(A0 , A1 , A2 ) be the set constructed in (6), T n C1 = n QA,F and 0 C2 = [ C(A, Fn , Fn+m ) ∪ n,m [ C(F0 , Fn , A). n Let C = C0 ∪ C1 ∪ C2 . If x ∈ [A]E \ C then there is some n for which [A/Fn ](x) 6= 0, so for all m we have [A/Fn+m ](x) ≥ [A/Fn ](x)[Fn /Fn+m ](x) ≥ 2m , and therefore [A/Fn ](x) → ∞. By the uniformity of the proofs of (6), (7) and Proposition A.3.7, C is ∆11 , so it remains to show that it admits a ∆11 compression. By the uniformity of the proofs of (6), (7) and Lemma A.3.2(2), it suffices to show that C1 \ (C0 ∪ C2 ) admits a ∆11 compression. First we show that C1 ∩ n QF0 n ,A admits a ∆11 compression. For this it suffices to show that C1 ∩ QF0 n ,A admits a ∆11 compression for all n (in a uniform way), by Lemma A.3.2(2). But by definition and E-invariance we have S Fn ∩ C1 ∩ QF0 n ,A ≺ A ∩ C1 ∩ QF0 n ,A ≺ Fn ∩ C1 ∩ QF0 n ,A , so Fn ∩ C1 ∩ QF0 n ,A admits a ∆11 compression, and since Fn is a complete section we are done by Lemma A.3.2(1). Next we consider C 0 = C1 \ (C0 ∪ C2 ∪ S Fn ,A ). For any x ∈ C 0 , n ∈ N, we have n Q0 [F0 /A](x) ≥ [F0 /Fn ](x)[Fn /A](x) ≥ 2n , so [F0 /A](x) = ∞ and x ∈ QF∞0 ,A . Thus C 0 ⊆ QF∞0 ,A admits a ∆11 compression. 0 ,B (9) This set is clearly ∆11 and it is E-invariant by (1). Next note that Y ⊆ [B]E \ QA ∞ S 0 ,B so we can decompose Y into Y0 = Y \ [A0 ]E and Y1 = Y ∩ [A0 ]E = n (Y ∩ QA ). n Since Y0 ∩ A0 = ∅ we clearly have Y0 ∩ A0  Y0 ∩ A1 , so it remains to show that 0 ,B 1 ,B Y1 ∩ A0  Y1 ∩ A1 . But by Lemma A.3.2(3) we have that A0 ∩ QA ∩ QA  m n A0 ,B A1 ,B A1 ∩ Qm ∩ Qn for m < n, so by Lemma A.3.2(2) we are done. 164 (E) Local measures. Proposition A.3.11. Let A ⊆ N be ∆11 . Then there is some C ∈ H such that lim n [A/Fn ](x) [N /Fn ](x) exists for x ∈ / C, and the limit is zero for x ∈ / [A]E ∪ C and is non-zero and finite for x ∈ [A]E \ C. Proof. Let C0 (A0 , A1 , A2 ), C1 , C2 (A) be the sets we have constructed in the proofs S of Lemma A.3.10(6)(7)(8), respectively, and take C = n,m C0 (A, Fn , Fn+m ) ∪ C1 ∪ S n C2 (A) ∪ n QA,F ∞ . By Lemma A.3.2(2) and the uniformity of Lemma A.3.10, C ∈ H . If x ∈ / [A]E ∪ C then [A/Fn ](x) = 0 and [N /Fn ](x) = 2n for all n, so the limit exists and is zero. Now suppose that x ∈ [A]E \ C. Then [Fn /Fn+m ](x) = 2m for all m, n ∈ N, and [A/Fn+m ](x) ≤ ([A/Fn ](x) + 1)([Fn /Fn+m ](x) + 1), so lim sup m→∞ [A/Fn+m ](x) [A/Fn ](x) + 1 ≤ . [N /Fn+m ](x) [N /Fn ](x) Thus the limit exists and is finite at x. To see that the limit is non-zero at x, note that [A/Fn+m ](x) ≥ [A/Fn ](x)[Fn /Fn+m ](x) for all m, n ∈ N, so [A/Fn+m ](x) [A/Fn ](x) ≥ m→∞ [N /F [N /Fn ](x) n+m ](x) lim inf for all n, and since [A/Fn ](x) → ∞ this lower bound must be non-zero for some n. Definition A.3.12. Let A ⊆ N be ∆11 and let CA ∈ H be the set constructed in the proof of Proposition A.3.11. We associate to A the local measure function m(A, ·) : N \ CA → R defined by m(A, x) = lim n [A/Fn ](x) . [N /Fn ](x) Note that the local measure function is ∆11 , uniformly in A. Lemma A.3.13. Let A, B, S ⊆ N be ∆11 . (1) If xEy then m(A, x) = m(A, y) for x, y ∈ / CA . 165 (2) Let Y = {x ∈ N \ (CA ∪ CB ) : m(A, x) < m(B, x)}. Then Y is ∆11 , E-invariant and A ∩ Y  B ∩ Y . (3) Suppose S is E-invariant. Then there is some C ∈ H such that for x ∈ / C, m(S, x) =   1 x ∈ S,  0 x∈ / S. (4) If S is E-invariant, then there is some C ∈ H such that for x ∈ S \ C we have m(A, x) = m(A ∩ S, x). Proof. (1) This follows from Lemma A.3.10(1). (2) This set is E-invariant by (1) and is ∆11 because the local measure functions are ∆11 . Now let Yn = {x ∈ Y : [A/Fn ](x) < [B/Fn ](x)}. The sets Yn are E-invariant, ∆11 and cover Y , so we have A ∩ Y  B ∩ Y by Lemma A.3.10(9) and Lemma A.3.2(2). (3) If x ∈ / S then [S/Fn ](x) = 0 for all n, so m(S, x) = 0. On the other hand, if x ∈ S S n n then [S/Fn ](x) = k ⇐⇒ x ∈ QS,F , so it suffices to show that k6=2n QS,F ∈H. k k This is done exactly as in the proof of Lemma A.3.10(7). (4) Let C0 (A, B, S) be the set constructed in the proof of Lemma A.3.10(4) and take S C = n C0 (A, Fn , S) ∪ CA ∪ CA∩S . Then C ∈ H by Lemma A.3.2(2) and clearly C works. Proposition A.3.14. Let A, B, S ⊆ N be ∆11 and let (An ) be a uniformly ∆11 sequence of subsets of N . (1) If A  B then there is some C ∈ H such that m(A, x) ≤ m(B, x) for x ∈ / C. (2) If A ∼ B then there is some C ∈ H such that m(A, x) = m(B, x) for x ∈ / C. (3) If A, B are disjoint then there is some C ∈ H such that m(A, x) + m(B, x) = m(A t B, x) for x ∈ / C. (4) Suppose the (An ) are pairwise disjoint, S is E-invariant and the partial maps P m(A, ·), m(An , ·) are defined on S. Suppose additionally that m(A, x) > n m(An , x) F for x ∈ S. Then there is some C ∈ H satisfying ( n An ) ∩ (S \ C)  A ∩ (S \ C). (5) If A = x∈ / C. F n An then there is some C ∈ H such that m(A, x) = P n m(An , x) for 166 Proof. (1) Let C = n C0 (A, B, Fn ) ∪ CA ∪ CB , where C0 (A0 , A1 , B) denotes the set constructed in the proof of Lemma A.3.10(2). S (2) This follows from (1). (3) Let C0 (A0 , A1 , B) and C1 be the sets we have constructed in the proofs of S Lemma A.3.10(5) and (7), respectively, and take C = n C0 (A, B, Fn ) ∪ CA ∪ CB ∪ C1 . (4) We construct recursively a sequence of ∆11 sets and functions Ãn , Bn , Cn , Sn , fn , gn such that Ãn+1 = Ãn \ Bn , Sn+1 = Sn \ Cn , fn : An ∩ Sn ∼ Bn ∩ Sn , gn : Cn ≺ Cn , P and m(Ãn , x) > k≥n m(Ak , x) for x ∈ Sn . To do this, we first set Ã0 = A, S0 = S. P Now suppose we have Ãn , Sn satisfying m(Ãn , x) > k≥n m(Ak , x) for x ∈ Sn . Then m(Ãn , x) > m(An , x) for x ∈ Sn , so by Lemma A.3.13(2) we can find Bn ⊆ Ãn and fn : An ∩ Sn ∼ Bn ∩ Sn . By (2), (3) and Lemma A.3.13(4) there are gn : Cn ≺ Cn such that for x ∈ Sn \ Cn we have m(An , x) = m(Bn , x) and m(Ãn , x) = m(Bn , x) + m(Ãn \ Bn , x). We then define Ãn+1 = Ãn \ Bn , Sn+1 = Sn \ Cn . By the uniformity of the proofs of (2), (3) and Lemma A.3.13, these sequences are S T uniformly ∆11 . Let C = n Cn , and note that S \ C = n Sn , so An ∩ (S \ C) ∼ Bn ∩ (S \ C) for all n. Thus by Lemma A.3.2(2) we have C ∈ H and ( G G An ) ∩ (S \ C) ∼ ( n Bn ) ∩ (S \ C) ⊆ A ∩ (S \ C). n (5) Let C0 (A, B), C1 (A, B) be the sets constructed in the proofs of (1) and (3), respectively, and let C̃ = CA ∪ [ [CAn ∪ C0 (A0 ∪ · · · ∪ An , A) ∪ C1 (A0 ∪ · · · ∪ An , An+1 )]. n Then for x ∈ / C̃ and n ∈ N we have X m(Ak , x) = m( k<n and therefore [ Ak , x) ≤ m(A, x), k<n / C̃. n m(An , x) ≤ m(A, x) for x ∈ P Now let C2 be the set constructed in the proof of Lemma A.3.10(7) and define C = C̃ ∪ C2 ∪ CN \A ∪ C1 (A, N \ A) ∪ [ n Then for x ∈ / C we have • n m(An , x) ≤ m(A, x), P [CFn ∪ CN \Fn ∪ C1 (Fn , N \ Fn )]. 167 • m(A, x) + m(N \ A, x) = m(N , x), • ∀n(m(Fn , x) = 2−n ), and • ∀n(m(Fn , x) + m(N \ Fn , x) = m(N , x)). Let Sk = {x ∈ / C : m(A, x) > n m(An , x) + 2−k }. These sets are ∆11 and ES P invariant, and if x ∈ / C ∪ k Sk then m(A, x) = n m(An , x). By the uniformity of the construction of C, Sk and Lemma A.3.2(2), it remains to show that each Sk ∈ H . P For x ∈ Sk we have m(N \ Fk , x) = m(A, x) + m(N \ A, x) − m(Fk , x) > m(N \ A, x) + X m(An , x). n By (4) there is some Ck ∈ H for which  Sk \ Ck =   [ An ∪ (N \ A) ∩ (Sk \ Ck )  (N \ Fk ) ∩ (Sk \ Ck ). n Since Fk is an E-complete section, this means that Sk \ Ck ∈ H , and hence that Sk ∈ H , as desired. (F) Proof of the Effective Nadkarni’s Theorem. Recall that we have fixed some sequence of maps (γn ) satisfying (1) of Theorem A.2.2. Fix now some τ, U, d, (Unk ) satisfying (2), (3) of Theorem A.2.2. Let CA be the set defined in Definition A.3.12, and let C0 (A, B), C1 (A, B), and C2 (A, (An )) be the sets constructed in the proofs of Proposition A.3.14 (2), (3), and (5), respectively. Now define C= [ {CU : U ∈ U} ∪ [ {C0 (U, γn U ) : U ∈ U , n ∈ N} ∪ [ {C1 (U, V \ U ) : U, V ∈ U} ∪ [ {C2 (U, (Unk )n ) : U ∈ U , k > 0}. By the uniformity of the constructions of the CA , C0 , C1 , C2 , along with the fact that U, (Unk ) are uniformly ∆11 , there is a uniformly ∆11 enumeration of the sets in this union, so C is ∆11 . By this uniformity and Lemma A.3.2(2), C admits a ∆11 compression. If N = C, then E admits a ∆11 compression. So suppose N 6= C and fix some x ∈ N \ C. By construction, the following hold for x: 168 • m(∅, x) = 0 and m(N , x) = 1; • for all U ∈ U , m(U, x) exists, is zero for x ∈ / [U ]E , and is non-zero and finite for x ∈ [U ]E ; • m(U, x) = m(γn U, x) for all U ∈ U , n ∈ N; • m(U t V, x) = m(U, x) + m(V, x) for all disjoint U, V ∈ U ; and • for all U ∈ U and k > 0, m(U, x) = k n m(Un , x). P Now define µ∗x (A) = inf{ X n m(Un , x) : Un ∈ U & A ⊆ [ Un }. n As in the classical proof of Nadkarni’s Theorem (cf. [BK96, p. 51-52] or [Slu, Theorem 2.8.1]), µ∗x is a metric outer measure whose restriction µx to the Borel sets is an E-invariant probability Borel measure satisfying µx (U ) = m(U, x), for U ∈ U. Thus, E admits an invariant probability Borel measure. A.4 A counterexample Let E be a ∆11 CBER on N . Nadkarni’s Theorem says that either E is compressible or E admits an invariant probability Borel measure. We have seen in Theorem A.1.4 that if E is compressible, then actually there is a ∆11 witness of this. On the other hand, if E is non-compressible, one may ask if there is an effective witness of this, i.e., if E admits a ∆11 invariant probability measure. It turns out that this is true if, for example, E is induced by a continuous, ∆11 action of a countable group on the Cantor space, but it is not true in general. Let P (C) denote the space of probability Borel measures on C. As with P (N ), we <N identify P (C) with the Π01 set of all ϕ ∈ [0, 1]2 satisfying ϕ(∅) = 1 and ϕ(s) = ϕ(s_ 0) + ϕ(s_ 1) for s ∈ 2<N . We then have the following: Proposition A.4.1. Let E be a CBER on the Cantor space C. Suppose there is a uniformly ∆11 sequence (fn ) of homeomorphisms of C inducing E, i.e., such that xEy ⇐⇒ ∃n(fn (x) = y). Then if E is non-compressible, E admits a ∆11 invariant probability measure. Proof. Let INVE ⊆ P (C) be the set of all E-invariant probability Borel measures on C. If E is non-compressible, then INVE is compact, ∆11 and non-empty. By the basis theorem [Mos09, 4F.11], INVE contains a ∆11 point, which is a ∆11 E-invariant probability measure on C. 169 Let E, F be CBERs on the standard Borel spaces X, Y respectively. We say that E is Borel invariantly embeddable to F , denoted E viB F , if there is an injective Borel map f : X → Y such that xEy ⇐⇒ f (x)F f (y), and such that additionally f (X) ⊆ Y is F -invariant. We say F is invariantly universal if E viB F for any CBER E. Clearly, all invariantly universal CBERs admit invariant probability Borel measures. Proposition A.4.2. There exists an invariantly universal ∆11 CBER on N that does not admit a ∆11 invariant probability measure. Proof. Let F∞ be the free group on a countably infinite set of generators, and take F0 to be the shift equivalence relation on N F∞ ∼ = N . Note that F0 is an invariantly universal ∆11 CBER. Let F1 be a compressible ∆11 CBER on N . Let T be an illfounded computable tree on N with no ∆11 branches (cf. [Mos09, 4D.10]), and define the equivalence relation E on N × N by (w, x)E(y, z) ⇐⇒ w = y & [(w ∈ [T ] & xF0 z) or (w ∈ / [T ] & xF1 z)]. Then E is a non-compressible invariantly universal ∆11 CBER on N × N ∼ = N , because T is ill-founded and F0 is non-compressible and invariantly universal. Now suppose for the sake of contradiction that E admits a ∆11 invariant probability measure µ. For s ∈ N<N , let Ns = {x ∈ N : s ⊆ x}, and define S = {s ∈ N<N : µ(Ns × N ) > 0}. Then S is a non-empty pruned ∆11 subtree of T , because if s ∈ / T then E|(Ns × N ) is compressible, so µ(Ns × N ) = 0. But then S, and hence T , has a ∆11 branch, a contradiction. Remark A.4.3. Let E be the equivalence relation induced by the shift action of F∞ on C F∞ , and let F r(C F∞ ) ⊆ C F∞ be the free part of C F∞ , i.e., the set of points x such that γx 6= x, ∀γ ∈ F∞ , γ 6= 1. Then E|F r(C F∞ ) is invariantly universal for CBERs that can be induced by a free Borel action of F∞ . Using the representation of ∆11 CBERs constructed in Section A.2, and [Mos09, 4F.14], one sees that the proof of [FKSV23, Theorem 3.3.1] is effective. In particular, there is a ∆11 , compact, E-invariant set K ⊆ C F∞ admitting a ∆11 isomorphism E|K ∼ = E|F r(C F∞ ). Now consider the equivalence relation F on N × C F∞ given by (w, x)F (y, z) ⇐⇒ w = y & xEz. 170 Let T be the tree from the proof of Proposition A.4.2 and let X = [T ] × F r(C F∞ ). Then F |X is invariantly universal for CBERs that can be induced by a free action of F∞ , so there is a Borel isomorphism F |X ∼ = E|F r(C F∞ ), and F |X does not admit a ∆11 invariant probability Borel measure. It follows that F |X is Borel isomorphic to a ∆11 compact subshift of C F∞ . However, by the proof of Proposition A.4.1, every such subshift admits a ∆11 invariant probability Borel measure, so there is no ∆11 isomorphism between F |X and a ∆11 compact subshift of C F∞ . In particular, F |X is a concrete witness to [FKSV23, Proposition 3.8.15]. A.5 Proof of Effective Ergodic Decomposition As noted in [Nad90], the proof of Nadkarni’s Theorem can be used to provide a proof of the Ergodic Decomposition Theorem (see also [Slu, Section 2.9]). We will now show that this argument can also be effectivized, providing a proof of the Effective Ergodic Decomposition Theorem for invariant measures from the proof of the Effective Nadkarni’s Theorem. This provides a different proof of a special case of Ditzen’s Effective Ergodic Decomposition Theorem [Dit92], which is proved more generally for quasi-invariant measures. Let E be a non-compressible CBER on the Baire space N , in order to prove the Ergodic Decomposition Theorem for E. We may partition N = X t Y into ∆11 E-invariant pieces so that E|X is aperiodic and every E|Y -class C ⊆ Y is finite. It is easy to see that the Ergodic Decomposition Theorem holds for E|Y , so we may assume that E is aperiodic. Fix (fn ), τ, U, d, (Unk ) satisfying Theorem A.2.2 for E. By the proof of the Effective Nadkarni’s Theorem, there is a ∆11 E-invariant set C ⊆ N and a local measure function m, such that that C admits a ∆11 compression and for each x ∈ N \ C there is a (unique) E-invariant probability Borel measure µx on X satisfying µx (U ) = m(U, x) for all U ∈ U . For ∆11 sets A, B ⊆ N , let QA,B be the associated decomposition (cf. Notation A.3.8). n 1 Let Fn be the uniformly ∆1 fundamental sequence for E used in the proof of the Effective Nadkarni’s Theorem, and for s ∈ N<N , let Ns = {x ∈ N : s ⊆ x}. For s ∈ N<N , n, k ∈ N define Ss,n,k =   (N \ [Ns ]E ) ∪ QNs ,Fn k = 0,  QNs ,Fn otherwise. 0 k By the proof of Theorem A.2.2, we may assume that Ss,n,k ∈ U for all s, n, k. 171 Now let Z = N \ (C ∪ s,n,k C0 (Ss,n,k )), where C0 (S) is the set constructed in the proof of Lemma A.3.13(3). By the uniformity of this construction and Lemma A.3.2(2), N \ Z is ∆11 and admits a ∆11 compression. By invariance of the local measure function, the assignment x 7→ µx is E-invariant. Additionally, as noted in the introduction, <N we may identify P (N ) with the subspace of ϕ ∈ [0, 1]N satisfying ϕ(∅) = 1 and P ϕ(s) = n ϕ(s_ n), for s ∈ N<N . Then, by uniformity in A of the local measure <N function m(A, x), the assignment x 7→ µx defines a ∆11 map Z → INVE ⊆ [0, 1]N . S For x ∈ Z, let Sx = {y ∈ Z : µy = µx }. Lemma A.5.1. For any x ∈ Z, µx (Sx ) = 1. Proof. If x ∈ Ss,n,k , then by definition of Z, E-invariance of Ss,n,k and the fact that Ss,n,k ∈ U , we have µx (Ss,n,k ) = m(Ss,n,k , x) = 1. Now define S̃x = Z ∩ {Ss,n,k : x ∈ Ss,n,k }. Since N \ Z is compressible, µx (Z) = 1, and so µx (S̃x ) = 1. If y ∈ S̃x , then [Ns /Fn ](x) = [Ns /Fn ](y) for all s, n, so µy (Ns ) = m(Ns , y) = m(Ns , x) = µx for all s ∈ N<N , and hence µy = µx . Therefore S̃x ⊆ Sx , and µx (Sx ) = 1. T Lemma A.5.2. Let S ⊆ N be E-invariant and Borel. Then there is an E-invariant compressible Borel set C ⊆ N such that for x ∈ / C we have µx (S) = m(S, x) =   1 x ∈ S,  0 x∈ / S. Proof. By relativizing, we may assume S is ∆11 . Repeat the proofs of this section, assuming this time that S ∈ U, to get a ∆11 set Z 0 ⊆ N and a ∆11 assignment Z 0 3 x 7→ µ0x ∈ INVE induced by a local measure function m0 . Note that m = m0 by uniformity of the construction of the local measure function, and hence µx = µ0x for x ∈ Z ∩ Z 0. Let C = (N \ Z ∩ Z 0 ) ∪ C0 (S), where C0 (S) is the set constructed in the proof of Lemma A.3.13(3). Then C admits a ∆11 compression, and if x ∈ / C then µx (S) = µ0x (S) = m0 (S, x) =   1 x ∈ S,  0 x∈ / S. 172 Proposition A.5.3. For any x ∈ Z, µx is the unique E-ergodic invariant probability Borel measure on E|Sx . Moreover, every E-ergodic invariant probability Borel measure is equal to µx , for some x ∈ Z. Proof. Fix x ∈ Z. Note that Sx is E-invariant, Borel and non-compressible (as it supports the E-invariant measure µx ). Now let Y ⊆ N be E-invariant and Borel. By Lemma A.5.2 there is an E-invariant compressible Borel set C ⊆ N such that for y ∈ / C, µy (Y ) ∈ {0, 1}. Since Sx is E-invariant and non-compressible, there must be some y ∈ Sx \ C. Then µx (Y ) = µy (Y ) ∈ {0, 1}. Since Y was arbitrary, µx is E-ergodic. Now let ν be any E-ergodic invariant probability Borel measure. For every s ∈ N<N , n ∈ N, there is a unique k(s, n) ∈ N such that ν(Ss,n,k(s,n) ) = 1. Define T S = s,n Ss,n,k(s,n) . Then ν(S) = 1, so in particular S is non-compressible, and hence S ∩ Z 6= ∅. Let x ∈ S ∩ Z. We claim that µx = ν. To see this, fix some s ∈ N<N , in order to show that µx (Ns ) = ν(Ns ). Note that [Ns /Fn ](x) = k(s, n), for all s, n, so that µx (Ns ) = limn k(s,n) (cf. 2n Definition A.3.9 and Definition A.3.12). We now consider two cases. If ν([Ns ]E ) = 0, then k(s, n) = 0 for all n, so µx (Ns ) = 0 = ν(Ns ). Now suppose ν([Ns ]E ) = 1. For all Ns ,Fn s ,Fn n, we have Ns ∩ QN k(s,n) ≈ k(s, n)(Fn ∩ Qk(s,n) ), so, as noted at the start of Section A.3, ν(Ns ) ∈ [k(s, n)2−n , (k(s, n) + 1)2−n ] for all n. Thus ν(Ns ) = lim n k(s, n) = µx (Ns ). 2n Finally, it remains to show that µx is the unique E-ergodic invariant probability Borel measure on E|Sx . To see this, let ν be any other such measure and write ν = µy for some y ∈ Z. Then ν(Sy ) = µy (Sy ) = 1, so ν(Sx ∩ Sy ) = 1. Thus Sx ∩ Sy 6= ∅, and so µx = µy = ν. Proposition A.5.4. Let µ, ν ∈ INVE . If µ(S) = ν(S) for all E-invariant Borel sets S ⊆ N , then µ = ν. Proof. Let A ⊆ N be ∆11 . As in the proof of Proposition A.5.3, we have n n n µ(A ∩ QA,F ) ∈ [k2−n µ(QA,F ), (k + 1)2−n µ(QA,F )]. k k k Similarly, n n n ν(A ∩ QA,F ) ∈ [k2−n ν(QA,F ), (k + 1)2−n ν(QA,F )]. k k k 173 n n n Since the sets QA,F are E-invariant, we have µ(QA,F ) = ν(QA,F ), and therefore k k k n n n |µ(A ∩ QA,F ) − ν(A ∩ QA,F )| ≤ 2−n µ(QA,F ). k k k It follows that |µ(A) − ν(A)| ≤ X n n |µ(A ∩ QA,F ) − ν(A ∩ QA,F )| ≤ 2−n k k k X n µ(QA,F ) ≤ 2−n . k k Since n was arbitrary, µ(A) = ν(A). Proposition A.5.5. For any ν ∈ INVE , ν = µx dν(x). R Proof. Let A ⊆ N be E-invariant. Then R Proposition A.5.4, ν = µx dν(x). R µx (A)dν(x) = ν(A ∩ Z) = ν(A). Thus, by Acknowledgements. We would like to thank B. Miller, F. Shinko, and Z. Vidnyánszky for many helpful discussions on this subject. Funding. This work was partially supported by NSF Grant DMS-1950475. 174 BIBLIOGRAPHY [AK00] Scot Adams and Alexander S. Kechris. Linear algebraic groups and countable Borel equivalence relations. J. Amer. Math. Soc. 13(4) (2000), 909–943. doi: 10.1090/S0894-0347-00-00341-6. 76, 102, 105. [Alp22] Andrei Alpeev. The invariant random order extension property is equivalent to amenability. 2022. arXiv: 2206.14177 [math.DS]. 14–16, 30, 44, 50. [BC24] Rishi Banerjee and Ruiyuan Chen. Structurable equivalence relations and Lω1 ω interpretations. 2024. arXiv: 2409.02896 [math.LO]. 12–14, 25, 55. [BK96] Howard Becker and Alexander S. Kechris. The Descriptive Set Theory of Polish Group Actions. 1st ed. Cambridge University Press, 1996. doi: 10.1017/CBO9780511735264. 24, 151, 156, 168. [Ber23] Anton Bernshteyn. Distributed algorithms, the Lovász Local Lemma, and descriptive combinatorics. Invent. Math. 223 (2023), 495–542. doi: 10.1007/s00222-023-01188-3. 12. [BD25] Anton Bernshteyn and Abhishek Dhawan. Borel Vizing’s theorem for graphs of subexponential growth. Proc. Amer. Math. Soc. 153(1) (2025), 7–14. doi: 10.1090/proc/17048. 44, 61, 62. [Bol01] Béla Bollobás. Random Graphs. 2nd ed. Cambridge University Press, 2001. doi: 10.1017/CBO9780511814068. 50. [BCW24] Matthew Bowen, Clinton T. Conley, and Felix Weilacher. Measurable Regular Subgraphs. 2024. arXiv: 2408.09597 [math.LO]. 44, 63–65. [BKS22] Matthew Bowen, Gábor Kun, and Marcin Sabok. Perfect matchings in hyperfinite graphings. 2022. arXiv: 2106.01988 [math.CO]. 44, 63–68. [BPZ24] Matthew Bowen, Antoine Poulin, and Jenna Zomback. One-ended spanning trees and definable combinatorics. Trans. Amer. Math. Soc. 377(12) (2024), 8411–8431. doi: 10.1090/tran/9186. 44, 63–66, 68. [BW23] Matthew Bowen and Felix Weilacher. Definable Kőnig theorems. Proc. Amer. Math. Soc. 151(11) (2023), 4991–4996. doi: 10.1090/proc/ 16355. 44, 61, 62. [BCGGRV22] Sebastian Brandt, Yi-Jun Chang, Jan Grebík, Christoph Grunau, Václav Rozhoň, and Zoltán Vidnyánszky. Local problems on trees from the perspectives of distributed algorithms, finitary factors, and descriptive combinatorics. Innovations in Theoretical Computer Science Conference (ITCS) 215 (2022), 29:1–29:26. doi: 10.4230/LIPIcs. ITCS.2022.29. 12. 175 [BM75] John Burgess and Douglas Miller. Remarks on invariant descriptive set theory. Fund. Math. 90(1) (1975), 53–75. doi: 10.4064/fm-90-153-75. 79. [CK18] Ruiyuan Chen and Alexander S. Kechris. Structurable equivalence relations. Fund. Math. 242(2) (2018), 109–185. doi: 10.4064/fm4287-2017. 2, 12, 13, 18, 26, 32, 34, 67. [CJMST20] Clinton T. Conley, Steve Jackson, Andrew S. Marks, Brandon Seward, and Robin D. Tucker-Drob. Hyperfiniteness and Borel combinatorics. J. Eur. Math. Soc. (JEMS) 22(3) (2020), 877–892. doi: 10.4171/ JEMS/935. 44, 61–63, 65. [CK13] Clinton T. Conley and Alexander S. Kechris. Measurable chromatic and independence numbers for ergodic graphs and group actions. Groups Geom. Dyn. 7(1) (2013), 127–180. doi: 10.4171/GGD/179. 44, 63, 65. [CM17] Clinton T. Conley and Benjamin D. Miller. Measurable Perfect Matchings for Acyclic Locally Countable Borel Graphs. J. Symb. Log. 82(1) (2017), 258–271. doi: 10.1017/jsl.2016.44. 44, 63, 65, 67. [CL17] Endre Csóka and Gabor Lippner. Invariant random perfect matchings in Cayley graphs. Groups Geom. Dyn. 11(1) (2017), 211–243. doi: 10.4171/GGD/395. 64. [dRM] Noé de Rancourt and Benjamin D. Miller. The Feldman-Moore, GlimmEffros, and Lusin-Novikov theorems over quotients. arXiv: 2105.05374 [math.LO]. Forthcoming. 75, 79, 110–112, 115, 118, 133–137, 142. [Dit92] Achim Ditzen. “Definable equivalence relations on Polish spaces”. PhD thesis. California Institute of Technology, 1992. doi: 10.7907/rf5sm868. 151–153, 156, 170. [DJK94] Randall Dougherty, Steve Jackson, and Alexander S. Kechris. The structure of hyperfinite Borel equivalence relations. Trans. Amer. Math. Soc. 341(1) (1994), 193–225. doi: 10.1090/S0002-9947-19941149121-0. 42, 47, 83. [Dye59] Henry A. Dye. On Groups of Measure Preserving Transformations. I. Amer. J. Math. 81(1) (1959), 119–159. doi: 10.2307/2372852. 31. [Far62] Roger H. Farrell. Representation of invariant measures. Illinois J. Math. 6(3) (1962), 447–467. doi: 10.1215/ijm/1255632504. 151, 152. [FM77] Jacob Feldman and Calvin C. Moore. Ergodic equivalence relations, cohomology, and von Neumann algebras. I. Trans. Amer. Math. Soc. 234(2) (1977), 289–324. doi: 10.1090/S0002-9947-1977-0578656-4. 22, 153. 176 [Fri81] Harvey Friedman. On the necessary use of abstract set theory. Adv. Math. 41(3) (1981), 209–280. doi: 10.1016/0001-8708(81)90021-9. 83. [FKSV23] Joshua R. Frisch, Alexander S. Kechris, Forte Shinko, and Zoltán Vidnyánszky. Realizations of countable Borel equivalence relations. 2023. arXiv: 2109.12486 [math.LO]. 11, 169, 170. [Gao08] Su Gao. Invariant descriptive set theory. Monographs and Textbooks in Pure and Applied Mathematics 293. Boca Raton, Florida: CRC Press, 2008. 2. [GJKS24] Su Gao, Steve Jackson, Edward Krohne, and Brandon Seward. Borel combinatorics of abelian group actions. 2024. arXiv: 2401 . 13866 [math.LO]. 64. [GX24] Su Gao and Ming Xiao. The Borel Ramsey properties for countable Borel equivalence relations. 2024. arXiv: 2401.14605 [math.LO]. 2, 12, 44, 48, 68. [GLM24] Yair Glasner, Yuqing Frank Lin, and Tom Meyerovitch. Extensions of invariant random orders on groups. Groups Geom. Dyn. 18(4) (2024), 1377–1401. doi: 10.4171/GGD/785. 14, 16, 50. [GMP17] Łukasz Grabowski, András Máthé, and Oleg Pikhurko. Measurable circle squaring. Annals of Mathematics 185(2) (2017), 671–710. doi: 10.4007/annals.2017.185.2.6. 64. [Gre25] Jan Grebík. Measurable Vizing’s theorem. Forum Math. Sigma 13 (2025), 1–24. doi: doi:10.1017/fms.2024.83. 61, 62. [GP20] Jan Grebík and Oleg Pikhurko. Measurable versions of Vizing’s theorem. Adv. Math. 374(18) (2020). doi: 10.1016/j.aim.2020.107378. 44, 61, 62. [GR23] Jan Grebík and Václav Rozhoň. Local problems on grids from the perspective of distributed algorithms, finitary factors, and descriptive combinatorics. Adv. Math. 431 (2023), 1–56. doi: 10.1016/j.aim. 2023.109241. 12. [HKL90] Leo A. Harrington, Alexander S. Kechris, and Alain Louveau. A Glimm-Effros dichotomy for Borel equivalence relations. J. Amer. Math. Soc. 3(4) (1990), 903–928. doi: 10.1090/S0894-0347-19901057041-5. 82–85, 150. [Hjo05] Greg Hjorth. A dichotomy theorem for being essentially countable. In Contemporary Mathematics. Ed. by Andreas Blass and Yi Zhang. Vol. 380. 2005, 109–127. doi: 10.1090/conm/380. 109. [HK01] Greg Hjorth and Alexander S. Kechris. Recent developments in the theory of Borel reducibility. Fund. Math. 170(1) (2001), 21–52. doi: 10.4064/fm170-1-2. 106. 177 [Hod93] Wilfrid Hodges. Model Theory. Cambridge University Press, 1993. doi: 10.1017/CBO9780511551574. 19, 34. [HP24] Tom Hutchcroft and Minghao Pan. Dimension jump at the uniqueness threshold for percolation in ∞+d dimensions. 2024. arXiv: 2412.15895 [math.PR]. 59. [JKL02] Steve Jackson, Alexander S. Kechris, and Alain Louveau. Countable Borel equivalence relations. J. Math. Log. 02(1) (2002), 1–80. doi: 10.1142/S0219061302000138. 62, 65. [Kan08] Vladimir G. Kanoveĭ. Borel equivalence relations: structure and classification. University Lecture Series 44. Providence, R.I.: American Mathematical Society, 2008. 2. [KL23] Alexander Kastner and Clark Lyons. Baire measurable matchings in non-amenable graphs. 2023. arXiv: 2310.20047 [math.LO]. 44, 64, 65. [Kec95] Alexander S. Kechris. Classical descriptive set theory. Graduate texts in mathematics 156. New York: Springer-Verlag, 1995. 402 pp. doi: 10.1007/978-1-4612-4190-4. 5, 18, 25, 31, 33, 37, 41, 42, 66, 67, 72, 78–85, 87, 88, 91, 103–105, 123, 129, 132, 152, 155. [Kec03] Alexander S. Kechris. “Actions of Polish Groups and Classification Problems”. In. Analysis and Logic. Cambridge University Press, 2003, 115–187. 85, 86. [Kec25] Alexander S. Kechris. The theory of countable Borel equivalence relations. Cambridge University Press, 2025. doi: 10.1017/9781009562256. 2, 13, 16, 18, 23, 31, 46, 51, 57, 61, 66, 107, 109. [KM20] Alexander S. Kechris and Andrew S. Marks. Descriptive Graph Combinatorics. 2020. url: https://pma.caltech.edu/documents/5616/ combinatorics20book.pdf (04/06/2025). 2, 12, 14, 56, 64. [KM04] Alexander S. Kechris and Benjamin D. Miller. Topics in Orbit Equivalence. Vol. 1852. Lecture Notes in Mathematics. Berlin, Heidelberg: Springer Berlin Heidelberg, 2004. doi: 10.1007/b99421. 23, 37, 42, 43, 45, 47. [KST99] Alexander S. Kechris, Sławomir Solecki, and Stevo Todorčević. Borel Chromatic Numbers. Adv. Math. 141(1) (1999), 1–44. doi: 10.1006/ aima.1998.1771. 16, 44, 45, 49, 55, 61, 93, 143, 146, 147. [Kun24] Gábor Kun. The measurable Hall theorem fails for treeings. 2024. arXiv: 2106.02013 [math.CO]. 44, 63. [Lac88] Miklós Laczkovich. Closed sets without measurable matching. Proc. Amer. Math. Soc. 103(3) (1988), 894–896. doi: 10.1090/S0002-99391988-0947676-2. 44, 63, 65. 178 [Lec09] Dominique Lecomte. A dichotomy characterizing analytic digraphs of uncountable Borel chromatic number in any dimension. Trans. Amer. Math. Soc. 361(8) (2009), 4181–4193. doi: 10.1090/S0002-9947-0904792-8. 6, 75, 89, 91, 93. [LM08] Dominique Lecomte and Benjamin D. Miller. Basis theorems for non-potentially closed sets and graphs of uncountable Borel chromatic number. J. Math. Log. 8(2) (2008), 121–162. doi: 10.1142/ S0219061308000749. 142. [LN11] Russell Lyons and Fedor Nazarov. Perfect matchings as IID factors on non-amenable groups. European J. Combin. 32(7) (2011), 1115–1125. doi: 10.1016/j.ejc.2011.03.008. 44, 63, 64. [LS99] Russell Lyons and Oded Schramm. Indistinguishability of Percolation Clusters. Ann. Probab. 27(4) (1999), 1809–1836. doi: 10.1214/aop/ 1022677549. 59, 68. [Mar16] Andrew S. Marks. A determinacy approach to Borel combinatorics. J. Amer. Math. Soc. 29(2) (2016), 579–600. doi: 10.1090/jams/836. 61, 63. [MU16] Andrew S. Marks and Spencer T. Unger. Baire measurable paradoxical decompositions via matchings. Adv. Math. 289 (2016), 397–410. doi: 10.1016/j.aim.2015.11.034. 44, 63–66. [Mila] B. D. Miller. “Essential values of cocycles and the Borel structure of R/Q”. url: https://glimmeffros.github.io/recent/cocycles. pdf (10/14/2024). 8, 113, 123, 127, 128, 132–134. [Mil11] Benjamin D. Miller. Dichotomy theorems for countably infinite dimensional analytic hypergraphs. Ann. Pure Appl. Logic 162(7) (2011), 561–565. doi: 10.1016/j.apal.2011.01.005. 93. [Mil12] Benjamin D. Miller. The graph-theoretic approach to descriptive set theory. Bull. Symb. Log. 18(4) (2012), 554–575. doi: 10.2178/bsl. 1804030. 6, 10, 75, 89, 92, 93, 122, 140. [Mil22] Benjamin D. Miller. A generalization of the G0 dichotomy and a strengthening of the EN0 dichotomy. J. Math. Log. 22(01) (2022). doi: 10.1142/S0219061321500288. 141. [Milb] Benjamin D. Miller. “Forceless, ineffective, powerless proofs of descriptive dichotomy theorems, lecture III”. url: https://glimmeffros. github.io/unpublished/paristhree.pdf (02/22/2024). 92, 100, 101. [Milc] Benjamin D. Miller. “Forceless, ineffective, powerless proofs of descriptive dichotomy theorems, lecture IV”. url: https://glimmeffros. github.io/unpublished/parisfour.pdf (02/22/2024). 93, 94. 179 [Mild] Benjamin D. Miller. “Invariant uniformizations and quasi-transversals”. url: https://glimmeffros.github.io/recent/uniformization. pdf (02/22/2024). 6, 74, 75, 79, 90, 111, 112, 118. [Mos09] Yiannis N. Moschovakis. Descriptive set theory. 2nd ed. Mathematical surveys and monographs 155. Providence, R.I.: American Mathematical Society, 2009. 82, 84, 102, 103, 127, 128, 147, 168, 169. [Mye76] Dale Myers. Invariant uniformization. Fund. Math. 91(1) (1976), 65– 72. doi: 10.4064/fm-91-1-65-72. 79, 80. [Nad90] Mahendra G. Nadkarni. On the existence of a finite invariant measure. Proc. Indian Acad. Sci. (Math. Sci.) 100(3) (1990), 203–220. doi: 10.1007/BF02837844. 24, 151, 152, 170. [OW80] Donald S. Ornstein and Benjamin Weiss. Ergodic theory of amenable group actions. I: The Rohlin lemma. Bull. Amer. Math. Soc. 2(1) (1980), 161–164. doi: 10.1090/S0273-0979-1980-14702-3. 31. [Paw15] Janusz Pawlikowski. Selivanovski hard sets are hard. Fund. Math. 228(1) (2015), 17–25. doi: 10.4064/fm228-1-2. 103. [Phe01] Robert R. Phelps. Lectures on Choquet’s theorem. Lecture Notes in Mathematics 1757. Berlin: Springer-Verlag, 2001. 130 pp. doi: 10.1007/b76887. 67. [Pik21] Oleg Pikhurko. “Borel combinatorics of locally finite graphs”. In. Surveys in Combinatorics 2021. Ed. by Konrad K. Dabrowski, Maximilien Gadouleau, Nicholas Georgiou, Matthew Johnson, George B. Mertzios, and Daniël Paulusma. London Mathematical Society Lecture Note Series. Cambridge University Press, 2021, 267–320. 2, 12. [QW22] Long Qian and Felix Weilacher. Descriptive Combinatorics, Computable Combinatorics, and ASI Algorithms. 2022. arXiv: 2206.08426 [math.LO]. 63. [Sab12] Marcin Sabok. Complexity of Ramsey null sets. Adv. Math. 230(3) (2012), 1184–1195. doi: 10.1016/j.aim.2012.03.001. 103, 105. [Sim09] Stephen G. Simpson. Subsystems of second order arithmetic. Perspectives in Logic. Cambridge: Cambridge University Press, 2009. 444 pp. doi: 10.1017/CBO9780511581007. 67. [Slu] Konstantin Slutsky. “Countable Borel equivalence relations”. url: https://kslutsky.com/lecture-notes/cber.pdf (12/15/2023). 153, 156, 157, 168, 170. [Sta85] Lee J. Stanley. Borel diagonalization and abstract set theory: Recent results of Harvey Friedman. In Harvey Friedman’s research on the foundations of mathematics. Ed. by Leo A. Harrington, Michael D. Morley, Andrej Scedrov, and Stephen G. Simpson. North-Holland, 1985, 11–86. doi: 10.1016/S0049-237X(09)70155-7. 108. 180 [Ste89] J. Michael Steele. Kingman’s subadditive ergodic theorem. Annales de l’I.H.P. Probabilités et statistiques 25(1) (1989), 93–98. 70. [Tho24] Riley Thornton. ∆11 effectivization in Borel combinatorics. J. Symb. Log. (2024), 1–24. doi: 10.1017/jsl.2024.38. Forthcoming. 151. [Tim23] Ádám Timár. A Factor Matching of Optimal Tail Between Poisson Processes. Combinatorica 43 (2023), 421–427. doi: 10.1007/s00493023-00051-6. 64, 68. [TV21] Stevo Todorčević and Zoltán Vidnyánszky. A complexity problem for Borel graphs. Invent. Math. 226(1) (2021), 225–249. doi: 10.1007/ s00222-021-01047-z. 76. [Var63] Veeravalli S. Varadarajan. Groups of automorphisms of Borel spaces. Trans. Amer. Math. Soc. 109 (1963), 191–220. doi: 10.1090/S00029947-1963-0159923-5. 151, 152. [Wei24] Felix Weilacher. Borel edge colorings for finite-dimensional groups. Israel J. Math. 263 (2024), 737–780. doi: 10.1007/s11856-0242640-y. 64.